negating both the hypothesis and conclusion of a conditional statement

Calcworkshop

Conditional Statement If Then's Defined in Geometry - 15+ Examples!

// Last Updated: January 21, 2020 - Watch Video //

In today’s geometry lesson , you’re going to learn all about conditional statements!

Jenn, Founder Calcworkshop ® , 15+ Years Experience (Licensed & Certified Teacher)

We’re going to walk through several examples to ensure you know what you’re doing.

In addition, this lesson will prepare you for deductive reasoning and two column proofs later on.

Here we go!

What are Conditional Statements?

To better understand deductive reasoning, we must first learn about conditional statements.

A conditional statement has two parts: hypothesis ( if ) and conclusion ( then ).

In fact, conditional statements are nothing more than “If-Then” statements!

Sometimes a picture helps form our hypothesis or conclusion. Therefore, we sometimes use Venn Diagrams to visually represent our findings and aid us in creating conditional statements.

But to verify statements are correct, we take a deeper look at our if-then statements. This is why we form the converse , inverse , and contrapositive of our conditional statements.

What is the Converse of a Statement?

Well, the converse is when we switch or interchange our hypothesis and conclusion.

Conditional Statement : “If today is Wednesday, then yesterday was Tuesday.”

Hypothesis : “If today is Wednesday” so our conclusion must follow “Then yesterday was Tuesday.”

So the converse is found by rearranging the hypothesis and conclusion, as Math Planet accurately states.

Converse : “If yesterday was Tuesday, then today is Wednesday.”

What is the Inverse of a Statement?

Now the inverse of an If-Then statement is found by negating (making negative) both the hypothesis and conclusion of the conditional statement.

So using our current conditional statement, “If today is Wednesday, then yesterday was Tuesday”.

Inverse : “If today is not Wednesday, then yesterday was not Tuesday.”

What is a Contrapositive?

And the contrapositive is formed by interchanging the hypothesis and conclusion and then negating both.

Contrapositive : “If yesterday was not Tuesday, then today is not Wednesday”

What is a Biconditional Statement?

A statement written in “if and only if” form combines a reversible statement and its true converse. In other words the conditional statement and converse are both true.

Continuing with our initial condition, “If today is Wednesday, then yesterday was Tuesday.”

Biconditional : “Today is Wednesday if and only if yesterday was Tuesday.”

Examples of Conditional Statements

In the video below we will look at several harder examples of how to form a proper statement, converse, inverse, and contrapositive. And here’s a big hint…

Whenever you see “con” that means you switch! It’s like being a con-artist!

Moreover, we will detail the process for coming up with reasons for our conclusions using known postulates. We will review the ten postulates that we have learned so far, and add a few more problems dealing with perpendicular lines, planes, and perpendicular bisectors.

After this lesson, we will be ready to tackle deductive reasoning head-on, and feel confident as we march onward toward learning two-column proofs!

Conditional Statements – Lesson & Examples (Video)

• Introduction to conditional statements
• 00:00:25 – What are conditional statements, converses, and biconditional statements? (Examples #1-2)
• 00:05:21 – Understanding venn diagrams (Examples #3-4)
• 00:11:07 – Supply the missing venn diagram and conditional statement for each question (Examples #5-8)
• Exclusive Content for Member’s Only
• 00:17:48 – Write the statement and converse then determine if they are reversible (Examples #9-12)
• 00:29:17 – Understanding the inverse, contrapositive, and symbol notation
• 00:35:33 – Write the statement, converse, inverse, contrapositive, and biconditional statements for each question (Examples #13-14)
• 00:45:40 – Using geometry postulates to verify statements (Example #15)
• 00:53:23 – What are perpendicular lines, perpendicular planes and the perpendicular bisector?
• 00:56:26 – Using the figure, determine if the statement is true or false (Example #16)
• Practice Problems with Step-by-Step Solutions
• Chapter Tests with Video Solutions

Get access to all the courses and over 450 HD videos with your subscription

Monthly and Yearly Plans Available

Get My Subscription Now

Still wondering if CalcWorkshop is right for you? Take a Tour and find out how a membership can take the struggle out of learning math.

Converse, Inverse & Contrapositive of Conditional Statement

Converse, inverse, and contrapositive of a conditional statement.

What we want to achieve in this lesson is to be familiar with the fundamental rules on how to convert or rewrite a conditional statement into its converse, inverse, and contrapositive. 

But first, we need to review what a conditional statement is because it is the foundation or precursor of the three related sentences that we are going to discuss in this lesson.

What is a Conditional Statement?

A conditional statement takes the form “If [latex]p[/latex], then [latex]q[/latex]” where [latex]p[/latex] is the hypothesis while [latex]q[/latex] is the conclusion. A conditional statement is also known as an implication .

Sometimes you may encounter (from other textbooks or resources) the words “antecedent” for the hypothesis and “consequent” for the conclusion. Don’t worry, they mean the same thing.

In addition, the statement “If [latex]p[/latex], then [latex]q[/latex]” is commonly written as the statement “[latex]p[/latex] implies [latex]q[/latex]” which is expressed symbolically as [latex]{\color{blue}p} \to {\color{red}q}[/latex].

Given a conditional statement, we can create related sentences namely: converse , inverse , and contrapositive . They are related sentences because they are all based on the original conditional statement.

Let’s go over each one of them!

The Converse of a Conditional Statement

For a given conditional statement [latex]{\color{blue}p} \to {\color{red}q}[/latex], we can write the converse statement by interchanging or swapping the roles of the hypothesis and conclusion of the original conditional statement. Therefore, the converse is the implication [latex]{\color{red}q} \to {\color{blue}p}[/latex].

Notice, the hypothesis [latex]\large{\color{blue}p}[/latex] of the conditional statement becomes the conclusion of the converse. On the other hand, the conclusion of the conditional statement [latex]\large{\color{red}p}[/latex] becomes the hypothesis of the converse.

The Inverse of a Conditional Statement

When you’re given a conditional statement [latex]{\color{blue}p} \to {\color{red}q}[/latex], the inverse statement is created by negating both the hypothesis and conclusion of the original conditional statement. Thus, the inverse is the implication ~[latex]\color{blue}p[/latex] [latex]\to[/latex] ~[latex]\color{red}q[/latex].

The symbol ~[latex]\color{blue}p[/latex] is read as “not [latex]p[/latex]” while ~[latex]\color{red}q[/latex] is read as “not [latex]q[/latex]” .

The Contrapositive of a Conditional Statement

Suppose you have the conditional statement [latex]{\color{blue}p} \to {\color{red}q}[/latex], we compose the contrapositive statement by interchanging the hypothesis and conclusion of the inverse of the same conditional statement.

In other words, to find the contrapositive, we first find the inverse of the given conditional statement then swap the roles of the hypothesis and conclusion. Therefore, the contrapositive of the conditional statement [latex]{\color{blue}p} \to {\color{red}q}[/latex] is the implication ~[latex]\color{red}q[/latex] [latex]\to[/latex] ~[latex]\color{blue}p[/latex].

Truth Tables of a Conditional Statement, and its Converse, Inverse, and Contrapositive

Now that we know how to symbolically write the converse, inverse, and contrapositive of a given conditional statement, it is time to state some interesting facts about these logical statements.

To save time, I have combined all the truth tables of a conditional statement, and its converse, inverse, and contrapositive into a single table .

Here are some of the important findings regarding the table above:

• The conditional statement is NOT logically equivalent to its converse and inverse.
• The conditional statement is logically equivalent to its contrapositive. Thus, [latex]{\color{blue}p} \to {\color{red}q}[/latex] [latex] \equiv [/latex] ~[latex]\color{red}q[/latex] [latex]\to[/latex] ~[latex]\color{blue}p[/latex].
• The converse is logically equivalent to the inverse of the original conditional statement. Therefore, [latex]{\color{red}q} \to {\color{blue}p}[/latex] [latex] \equiv [/latex] ~[latex]\color{blue}p[/latex] [latex]\to[/latex] ~[latex]\color{red}q[/latex].

You may also be interested in these related math lessons or tutorials:

Introduction to Truth Tables, Statements, and Logical Connectives

Truth Tables of Five (5) Common Logical Connectives or Operators

A free service from Mattecentrum

If-then statement

• Logical correct I
• Logical correct II

When we previously discussed inductive reasoning we based our reasoning on examples and on data from earlier events. If we instead use facts, rules and definitions then it's called deductive reasoning.

We will explain this by using an example.

If you get good grades then you will get into a good college.

The part after the "if": you get good grades - is called a hypotheses and the part after the "then" - you will get into a good college - is called a conclusion.

Hypotheses followed by a conclusion is called an If-then statement or a conditional statement.

This is noted as

$$p \to q$$

This is read - if p then q.

A conditional statement is false if hypothesis is true and the conclusion is false. The example above would be false if it said "if you get good grades then you will not get into a good college".

If we re-arrange a conditional statement or change parts of it then we have what is called a related conditional.

Our conditional statement is: if a population consists of 50% men then 50% of the population must be women.

If we exchange the position of the hypothesis and the conclusion we get a converse statemen t: if a population consists of 50% women then 50% of the population must be men.

$$q\rightarrow p$$

If both statements are true or if both statements are false then the converse is true. A conditional and its converse do not mean the same thing

If we negate both the hypothesis and the conclusion we get a inverse statemen t: if a population do not consist of 50% men then the population do not consist of 50% women.

$$\sim p\rightarrow \: \sim q$$

The inverse is not true juest because the conditional is true. The inverse always has the same truth value as the converse.

We could also negate a converse statement, this is called a contrapositive statemen t:  if a population do not consist of 50% women then the population do not consist of 50% men.

$$\sim q\rightarrow \: \sim p$$

The contrapositive does always have the same truth value as the conditional. If the conditional is true then the contrapositive is true.

A pattern of reaoning is a true assumption if it always lead to a true conclusion. The most common patterns of reasoning are detachment and syllogism.

If we turn of the water in the shower, then the water will stop pouring.

If we call the first part p and the second part q then we know that p results in q. This means that if p is true then q will also be true. This is called the law of detachment and is noted:

$$\left [ (p \to q)\wedge p \right ] \to q$$

The law of syllogism tells us that if p → q and q → r then p → r is also true.

This is noted:

$$\left [ (p \to q)\wedge (q \to r ) \right ] \to (p \to r)$$

If the following statements are true:

If we turn of the water (p), then the water will stop pouring (q). If the water stops pouring (q) then we don't get wet any more (r).

Then the law of syllogism tells us that if we turn of the water (p) then we don't get wet (r) must be true.

Video lesson

Write a converse, inverse and contrapositive to the conditional

"If you eat a whole pint of ice cream, then you won't be hungry"

• Angles, parallel lines and transversals
• Congruent triangles
• More about triangles
• Inequalities
• Mean and geometry
• The converse of the Pythagorean theorem and special triangles
• Properties of parallelograms
• Common types of transformation
• Transformation using matrices
• Basic information about circles
• Inscribed angles and polygons
• Advanced information about circles
• Parallelogram, triangles etc
• The surface area and the volume of pyramids, prisms, cylinders and cones
• SAT Overview
• ACT Overview

Conditional Statements

Andymath.com features free videos, notes, and practice problems with answers! Printable pages make math easy. Are you ready to be a mathmagician?

Practice Problems & Videos

\(\textbf{1)}\) “if a figure has 3 sides, then it is a triangle.” state the hypothesis. show answer the hypothesis is “a figure has 3 sides.”, \(\textbf{2)}\) “if a figure has 3 sides, then it is a triangle.” state the conclusion. show answer the conclusion is “a figure is a triangle.”, \(\textbf{3)}\) “if a figure has 3 sides, then it is a triangle.” state the converse. show answer the converse is “if a figure is a triangle, then it has 3 sides.”, \(\textbf{4)}\) “if a figure has 3 sides, then it is a triangle.” state the inverse. show answer the inverse is “if a figure does not have 3 sides, then it is not a triangle.”, \(\textbf{5)}\) “if a figure has 3 sides, then it is a triangle.” state the contrapositive. show answer the contrapositive is “if a figure is not a triangle, then it does not have 3 sides.”, \(\textbf{6)}\) “if a figure has 3 sides, then it is a triangle.” state the biconditional. show answer the biconditional is “a figure has 3 sides, if and only if, it is a triangle.”, challenge problems, \(\textbf{7)}\) create a venn diagram for “all circles are ellipses.” show answer one example below, \(\textbf{8)}\) create a venn diagram for “if you don’t have an ellipse, then you don’t have a circle.” show answer note it is the same answer as number 7. they are equivalent statements., \(\textbf{9)}\) write 2 conditional statements based on the venn diagram below. show answer “if a square, then a rectangle.” or “all squares are rectangles” and “if not a rectangle, not a square.” or “all non-rectangles are non-squares”, see related pages\(\), \(\bullet\text{ geometry homepage}\) \(\,\,\,\,\,\,\,\,\text{all the best topics…}\), \(\bullet\text{ law of syllogism}\) \(\,\,\,\,\,\,\,\,\text{if p then q,}\) \(\,\,\,\,\,\,\,\,\text{if q then r,}\) \(\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{if p then r…}\), \(\bullet\text{ law of detachment}\) \(\,\,\,\,\,\,\,\,\text{if p then q,}\) \(\,\,\,\,\,\,\,\,\text{p is true,}\) \(\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{q is true…}\), a conditional statement is a statement in the form “if p, then q,” where p and q are called the hypothesis and conclusion, respectively. the statement “if it is raining, then the ground is wet” is an example of a conditional statement. the converse of a conditional statement is formed by flipping the order in which the hypothesis and conclusion appear. for example, the converse of the statement “if it is raining, then the ground is wet” is “if the ground is wet, then it is raining.” the inverse of a conditional statement is formed by negating both the hypothesis and conclusion. for example, the inverse of the statement “if it is raining, then the ground is wet” is “if it is not raining, then the ground is not wet” the contrapositive of a conditional statement is formed by negating both the hypothesis and conclusion and flipping the order in which they appear. for example, the contrapositive of the statement “if it is raining, then the ground is wet” is “if the ground is not wet, then it is not raining.” a biconditional statement is a statement in the form “if and only if p, then q,” which is equivalent to the statement “p if and only if q.” this means that p and q are either both true or both false. for example, the statement “if and only if it is raining, the ground is wet” is a biconditional statement. in geometry class, students learn about conditional statements and their related concepts (inverse, converse, contrapositive, and biconditional) in order to make logical deductions about geometric figures and their properties. these concepts are often used to prove theorems and solve problems. andymath.com is a free math website with the mission of helping students, teachers and tutors find helpful notes, useful sample problems with answers including step by step solutions, and other related materials to supplement classroom learning. if you have any requests for additional content, please contact andy at [email protected] . he will promptly add the content. topics cover elementary math , middle school , algebra , geometry , algebra 2/pre-calculus/trig , calculus and probability/statistics . in the future, i hope to add physics and linear algebra content. visit me on youtube , tiktok , instagram and facebook . andymath content has a unique approach to presenting mathematics. the clear explanations, strong visuals mixed with dry humor regularly get millions of views. we are open to collaborations of all types, please contact andy at [email protected] for all enquiries. to offer financial support, visit my patreon page. let’s help students understand the math way of thinking thank you for visiting. how exciting.

Conditional Statement – Definition, Truth Table, Examples, FAQs

What is a conditional statement, how to write a conditional statement, what is a biconditional statement, solved examples on conditional statements, practice problems on conditional statements, frequently asked questions about conditional statements.

A conditional statement is a statement that is written in the “If p, then q” format. Here, the statement p is called the hypothesis and q is called the conclusion. It is a fundamental concept in logic and mathematics. 

Conditional statement symbol :  p → q

A conditional statement consists of two parts.

• The “if” clause, which presents a condition or hypothesis.
• The “then” clause, which indicates the consequence or result that follows if the condition is true. 

Example : If you brush your teeth, then you won’t get cavities.

Hypothesis (Condition): If you brush your teeth

Conclusion (Consequence): then you won’t get cavities 

Conditional Statement: Definition

A conditional statement is characterized by the presence of “if” as an antecedent and “then” as a consequent. A conditional statement, also known as an “if-then” statement consists of two parts:

• The “if” clause (hypothesis): This part presents a condition, situation, or assertion. It is the initial condition that is being considered.
• The “then” clause (conclusion): This part indicates the consequence, result, or action that will occur if the condition presented in the “if” clause is true or satisfied. 

Related Worksheets

Representation of Conditional Statement

The conditional statement of the form ‘If p, then q” is represented as p → q. 

It is pronounced as “p implies q.”

Different ways to express a conditional statement are:

• p implies q
• p is sufficient for q
• q is necessary for p

Parts of a Conditional Statement

There are two parts of conditional statements, hypothesis and conclusion. The hypothesis or condition will begin with the “if” part, and the conclusion or action will begin with the “then” part. A conditional statement is also called “implication.”

Conditional Statements Examples:

Example 1: If it is Sunday, then you can go to play. 

Hypothesis: If it is Sunday

Conclusion: then you can go to play. 

Example 2: If you eat all vegetables, then you can have the dessert.

Condition: If you eat all vegetables

Conclusion: then you can have the dessert 

To form a conditional statement, follow these concise steps:

Step 1 : Identify the condition (antecedent or “if” part) and the consequence (consequent or “then” part) of the statement.

Step 2 : Use the “if… then…” structure to connect the condition and consequence.

Step 3 : Ensure the statement expresses a logical relationship where the condition leads to the consequence.

Example 1 : “If you study (condition), then you will pass the exam (consequence).” 

This conditional statement asserts that studying leads to passing the exam. If you study (condition is true), then you will pass the exam (consequence is also true).

Example 2 : If you arrange the numbers from smallest to largest, then you will have an ascending order.

Hypothesis: If you arrange the numbers from smallest to largest

Conclusion: then you will have an ascending order

Truth Table for Conditional Statement

The truth table for a conditional statement is a table used in logic to explore the relationship between the truth values of two statements. It lists all possible combinations of truth values for “p” and “q” and determines whether the conditional statement is true or false for each combination. 

The truth value of p → q is false only when p is true and q is False. 

If the condition is false, the consequence doesn’t affect the truth of the conditional; it’s always true.

In all the other cases, it is true.

The truth table is helpful in the analysis of possible combinations of truth values for hypothesis or condition and conclusion or action. It is useful to understand the presence of truth or false statements. 

Converse, Inverse, and Contrapositive

The converse, inverse, and contrapositive are three related conditional statements that are derived from an original conditional statement “p → q.” 

Consider a conditional statement: If I run, then I feel great.

• Converse: 

The converse of “p → q” is “q → p.” It reverses the order of the original statement. While the original statement says “if p, then q,” the converse says “if q, then p.” 

Converse: If I feel great, then I run.

• Inverse: 

The inverse of “p → q” is “~p → ~q,” where “” denotes negation (opposite). It negates both the antecedent (p) and the consequent (q). So, if the original statement says “if p, then q,” the inverse says “if not p, then not q.”

Inverse : If I don’t run, then I don’t feel great.

• Contrapositive: 

The contrapositive of “p → q” is “~q → ~p.” It reverses the order and also negates both the statements. So, if the original statement says “if p, then q,” the contrapositive says “if not q, then not p.”

Contrapositive: If I don’t feel great, then I don’t run.

A biconditional statement is a type of compound statement in logic that expresses a bidirectional or two-way relationship between two statements. It asserts that “p” is true if and only if “q” is true, and vice versa. In symbolic notation, a biconditional statement is represented as “p ⟺ q.”

In simpler terms, a biconditional statement means that the truth of “p” and “q” are interdependent. 

If “p” is true, then “q” must also be true, and if “q” is true, then “p” must be true. Conversely, if “p” is false, then “q” must be false, and if “q” is false, then “p” must be false. 

Biconditional statements are often used to express equality, equivalence, or conditions where two statements are mutually dependent for their truth values. 

Examples : 

• I will stop my bike if and only if the traffic light is red.  
• I will stay if and only if you play my favorite song.

Facts about Conditional Statements

• The negation of a conditional statement “p → q” is expressed as “p and not q.” It is denoted as “𝑝 ∧ ∼𝑞.” 
• The conditional statement is not logically equivalent to its converse and inverse.
• The conditional statement is logically equivalent to its contrapositive. 
• Thus, we can write p → q ∼q → ∼p

In this article, we learned about the fundamentals of conditional statements in mathematical logic, including their structure, parts, truth tables, conditional logic examples, and various related concepts. Understanding conditional statements is key to logical reasoning and problem-solving. Now, let’s solve a few examples and practice MCQs for better comprehension.

Example 1: Identify the hypothesis and conclusion. 

If you sing, then I will dance.

Solution : 

Given statement: If you sing, then I will dance.

Here, the antecedent or the hypothesis is “if you sing.”

The conclusion is “then I will dance.”

Example 2: State the converse of the statement: “If the switch is off, then the machine won’t work.” 

Here, p: The switch is off

q: The machine won’t work.

The conditional statement can be denoted as p → q.

Converse of p → q is written by reversing the order of p and q in the original statement.

Converse of  p → q is q → p.

Converse of  p → q: q → p: If the machine won’t work, then the switch is off.

Example 3: What is the truth value of the given conditional statement? 

If 2+2=5 , then pigs can fly.

Solution:  

q: Pigs can fly.

The statement p is false. Now regardless of the truth value of statement q, the overall statement will be true. 

F → F = T

Hence, the truth value of the statement is true. 

Conditional Statement - Definition, Truth Table, Examples, FAQs

Attend this quiz & Test your knowledge.

What is the antecedent in the given conditional statement? If it’s sunny, then I’ll go to the beach.

A conditional statement can be expressed as, what is the converse of “a → b”, when the antecedent is true and the consequent is false, the conditional statement is.

What is the meaning of conditional statements?

Conditional statements, also known as “if-then” statements, express a cause-and-effect or logical relationship between two propositions.

When does the truth value of a conditional statement is F?

A conditional statement is considered false when the antecedent is true and the consequent is false.

What is the contrapositive of a conditional statement?

The contrapositive reverses the order of the statements and also negates both the statements. It is equivalent in truth value to the original statement.

RELATED POSTS

• Ordering Decimals: Definition, Types, Examples
• Decimal to Octal: Steps, Methods, Conversion Table
• Lattice Multiplication – Definition, Method, Examples, Facts, FAQs
• X Intercept – Definition, Formula, Graph, Examples
• Lateral Face – Definition With Examples

Math & ELA | PreK To Grade 5

Kids see fun., you see real learning outcomes..

Make study-time fun with 14,000+ games & activities, 450+ lesson plans, and more—free forever.

Parents, Try for Free Teachers, Use for Free

• school Campus Bookshelves
• menu_book Bookshelves
• perm_media Learning Objects
• login Login
• how_to_reg Request Instructor Account
• hub Instructor Commons

Margin Size

• Download Page (PDF)
• Download Full Book (PDF)
• Periodic Table
• Physics Constants
• Scientific Calculator
• Reference & Cite
• Tools expand_more
• Readability

selected template will load here

This action is not available.

1.1: Statements and Conditional Statements

• Last updated
• Save as PDF
• Page ID 7034

• Ted Sundstrom
• Grand Valley State University via ScholarWorks @Grand Valley State University

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)

( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)

\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)

\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

\( \newcommand{\Span}{\mathrm{span}}\)

\( \newcommand{\id}{\mathrm{id}}\)

\( \newcommand{\kernel}{\mathrm{null}\,}\)

\( \newcommand{\range}{\mathrm{range}\,}\)

\( \newcommand{\RealPart}{\mathrm{Re}}\)

\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

\( \newcommand{\Argument}{\mathrm{Arg}}\)

\( \newcommand{\norm}[1]{\| #1 \|}\)

\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)

\( \newcommand{\vectorA}[1]{\vec{#1}}      % arrow\)

\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}}      % arrow\)

\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

\( \newcommand{\vectorC}[1]{\textbf{#1}} \)

\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)

\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)

\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)

Much of our work in mathematics deals with statements. In mathematics, a statement is a declarative sentence that is either true or false but not both. A statement is sometimes called a proposition . The key is that there must be no ambiguity. To be a statement, a sentence must be true or false, and it cannot be both. So a sentence such as "The sky is beautiful" is not a statement since whether the sentence is true or not is a matter of opinion. A question such as "Is it raining?" is not a statement because it is a question and is not declaring or asserting that something is true.

Some sentences that are mathematical in nature often are not statements because we may not know precisely what a variable represents. For example, the equation 2\(x\)+5 = 10 is not a statement since we do not know what \(x\) represents. If we substitute a specific value for \(x\) (such as \(x\) = 3), then the resulting equation, 2\(\cdot\)3 +5 = 10 is a statement (which is a false statement). Following are some more examples:

• There exists a real number \(x\) such that 2\(x\)+5 = 10. This is a statement because either such a real number exists or such a real number does not exist. In this case, this is a true statement since such a real number does exist, namely \(x\) = 2.5.
• For each real number \(x\), \(2x +5 = 2 \left( x + \dfrac{5}{2}\right)\). This is a statement since either the sentence \(2x +5 = 2 \left( x + \dfrac{5}{2}\right)\) is true when any real number is substituted for \(x\) (in which case, the statement is true) or there is at least one real number that can be substituted for \(x\) and produce a false statement (in which case, the statement is false). In this case, the given statement is true.
• Solve the equation \(x^2 - 7x +10 =0\). This is not a statement since it is a directive. It does not assert that something is true.
• \((a+b)^2 = a^2+b^2\) is not a statement since it is not known what \(a\) and \(b\) represent. However, the sentence, “There exist real numbers \(a\) and \(b\) such that \((a+b)^2 = a^2+b^2\)" is a statement. In fact, this is a true statement since there are such integers. For example, if \(a=1\) and \(b=0\), then \((a+b)^2 = a^2+b^2\).
• Compare the statement in the previous item to the statement, “For all real numbers \(a\) and \(b\), \((a+b)^2 = a^2+b^2\)." This is a false statement since there are values for \(a\) and \(b\) for which \((a+b)^2 \ne a^2+b^2\). For example, if \(a=2\) and \(b=3\), then \((a+b)^2 = 5^2 = 25\) and \(a^2 + b^2 = 2^2 +3^2 = 13\).

Progress Check 1.1: Statements

Which of the following sentences are statements? Do not worry about determining whether a statement is true or false; just determine whether each sentence is a statement or not.

• 2\(\cdot\)7 + 8 = 22.
• \((x-1) = \sqrt(x + 11)\).
• \(2x + 5y = 7\).
• There are integers \(x\) and \(y\) such that \(2x + 5y = 7\).
• There are integers \(x\) and \(y\) such that \(23x + 27y = 52\).
• Given a line \(L\) and a point \(P\) not on that line, there is a unique line through \(P\) that does not intersect \(L\).
• \((a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\).
• \((a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\) for all real numbers \(a\) and \(b\).
• The derivative of \(f(x) = \sin x\) is \(f' (x) = \cos x\).
• Does the equation \(3x^2 - 5x - 7 = 0\) have two real number solutions?
• If \(ABC\) is a right triangle with right angle at vertex \(B\), and if \(D\) is the midpoint of the hypotenuse, then the line segment connecting vertex \(B\) to \(D\) is half the length of the hypotenuse.
• There do not exist three integers \(x\), \(y\), and \(z\) such that \(x^3 + y^2 = z^3\).

Add texts here. Do not delete this text first.

How Do We Decide If a Statement Is True or False?

In mathematics, we often establish that a statement is true by writing a mathematical proof. To establish that a statement is false, we often find a so-called counterexample. (These ideas will be explored later in this chapter.) So mathematicians must be able to discover and construct proofs. In addition, once the discovery has been made, the mathematician must be able to communicate this discovery to others who speak the language of mathematics. We will be dealing with these ideas throughout the text.

For now, we want to focus on what happens before we start a proof. One thing that mathematicians often do is to make a conjecture beforehand as to whether the statement is true or false. This is often done through exploration. The role of exploration in mathematics is often difficult because the goal is not to find a specific answer but simply to investigate. Following are some techniques of exploration that might be helpful.

Techniques of Exploration

• Guesswork and conjectures . Formulate and write down questions and conjectures. When we make a guess in mathematics, we usually call it a conjecture.

For example, if someone makes the conjecture that \(\sin(2x) = 2 \sin(x)\), for all real numbers \(x\), we can test this conjecture by substituting specific values for \(x\). One way to do this is to choose values of \(x\) for which \(\sin(x)\)is known. Using \(x = \frac{\pi}{4}\), we see that

\(\sin(2(\frac{\pi}{4})) = \sin(\frac{\pi}{2}) = 1,\) and

\(2\sin(\frac{\pi}{4}) = 2(\frac{\sqrt2}{2}) = \sqrt2\).

Since \(1 \ne \sqrt2\), these calculations show that this conjecture is false. However, if we do not find a counterexample for a conjecture, we usually cannot claim the conjecture is true. The best we can say is that our examples indicate the conjecture is true. As an example, consider the conjecture that

If \(x\) and \(y\) are odd integers, then \(x + y\) is an even integer.

We can do lots of calculation, such as \(3 + 7 = 10\) and \(5 + 11 = 16\), and find that every time we add two odd integers, the sum is an even integer. However, it is not possible to test every pair of odd integers, and so we can only say that the conjecture appears to be true. (We will prove that this statement is true in the next section.)

• Use of prior knowledge. This also is very important. We cannot start from square one every time we explore a statement. We must make use of our acquired mathematical knowledge. For the conjecture that \(\sin (2x) = 2 \sin(x)\), for all real numbers \(x\), we might recall that there are trigonometric identities called “double angle identities.” We may even remember the correct identity for \(\sin (2x)\), but if we do not, we can always look it up. We should recall (or find) that for all real numbers \(x\), \[\sin(2x) = 2 \sin(x)\cos(x).\]
• We could use this identity to argue that the conjecture “for all real numbers \(x\), \(\sin (2x) = 2 \sin(x)\)” is false, but if we do, it is still a good idea to give a specific counterexample as we did before.
• Cooperation and brainstorming . Working together is often more fruitful than working alone. When we work with someone else, we can compare notes and articulate our ideas. Thinking out loud is often a useful brainstorming method that helps generate new ideas.

Progress Check 1.2: Explorations

Use the techniques of exploration to investigate each of the following statements. Can you make a conjecture as to whether the statement is true or false? Can you determine whether it is true or false?

• \((a + b)^2 = a^2 + b^2\), for all real numbers a and b.
• There are integers \(x\) and \(y\) such that \(2x + 5y = 41\).
• If \(x\) is an even integer, then \(x^2\) is an even integer.
• If \(x\) and \(y\) are odd integers, then \(x \cdot y\) is an odd integer.

Conditional Statements

One of the most frequently used types of statements in mathematics is the so-called conditional statement. Given statements \(P\) and \(Q\), a statement of the form “If \(P\) then \(Q\)” is called a conditional statement . It seems reasonable that the truth value (true or false) of the conditional statement “If \(P\) then \(Q\)” depends on the truth values of \(P\) and \(Q\). The statement “If \(P\) then \(Q\)” means that \(Q\) must be true whenever \(P\) is true. The statement \(P\) is called the hypothesis of the conditional statement, and the statement \(Q\) is called the conclusion of the conditional statement. Since conditional statements are probably the most important type of statement in mathematics, we give a more formal definition.

A conditional statement is a statement that can be written in the form “If \(P\) then \(Q\),” where \(P\) and \(Q\) are sentences. For this conditional statement, \(P\) is called the hypothesis and \(Q\) is called the conclusion .

Intuitively, “If \(P\) then \(Q\)” means that \(Q\) must be true whenever \(P\) is true. Because conditional statements are used so often, a symbolic shorthand notation is used to represent the conditional statement “If \(P\) then \(Q\).” We will use the notation \(P \to Q\) to represent “If \(P\) then \(Q\).” When \(P\) and \(Q\) are statements, it seems reasonable that the truth value (true or false) of the conditional statement \(P \to Q\) depends on the truth values of \(P\) and \(Q\). There are four cases to consider:

• \(P\) is true and \(Q\) is true.
• \(P\) is false and \(Q\) is true.
• \(P\) is true and \(Q\) is false.
• \(P\) is false and \(Q\) is false.

The conditional statement \(P \to Q\) means that \(Q\) is true whenever \(P\) is true. It says nothing about the truth value of \(Q\) when \(P\) is false. Using this as a guide, we define the conditional statement \(P \to Q\) to be false only when \(P\) is true and \(Q\) is false, that is, only when the hypothesis is true and the conclusion is false. In all other cases, \(P \to Q\) is true. This is summarized in Table 1.1 , which is called a truth table for the conditional statement \(P \to Q\). (In Table 1.1 , T stands for “true” and F stands for “false.”)

Table 1.1: Truth Table for \(P \to Q\)

The important thing to remember is that the conditional statement \(P \to Q\) has its own truth value. It is either true or false (and not both). Its truth value depends on the truth values for \(P\) and \(Q\), but some find it a bit puzzling that the conditional statement is considered to be true when the hypothesis P is false. We will provide a justification for this through the use of an example.

Example 1.3:

Suppose that I say

“If it is not raining, then Daisy is riding her bike.”

We can represent this conditional statement as \(P \to Q\) where \(P\) is the statement, “It is not raining” and \(Q\) is the statement, “Daisy is riding her bike.”

Although it is not a perfect analogy, think of the statement \(P \to Q\) as being false to mean that I lied and think of the statement \(P \to Q\) as being true to mean that I did not lie. We will now check the truth value of \(P \to Q\) based on the truth values of \(P\) and \(Q\).

• Suppose that both \(P\) and \(Q\) are true. That is, it is not raining and Daisy is riding her bike. In this case, it seems reasonable to say that I told the truth and that\(P \to Q\) is true.
• Suppose that \(P\) is true and \(Q\) is false or that it is not raining and Daisy is not riding her bike. It would appear that by making the statement, “If it is not raining, then Daisy is riding her bike,” that I have not told the truth. So in this case, the statement \(P \to Q\) is false.
• Now suppose that \(P\) is false and \(Q\) is true or that it is raining and Daisy is riding her bike. Did I make a false statement by stating that if it is not raining, then Daisy is riding her bike? The key is that I did not make any statement about what would happen if it was raining, and so I did not tell a lie. So we consider the conditional statement, “If it is not raining, then Daisy is riding her bike,” to be true in the case where it is raining and Daisy is riding her bike.
• Finally, suppose that both \(P\) and \(Q\) are false. That is, it is raining and Daisy is not riding her bike. As in the previous situation, since my statement was \(P \to Q\), I made no claim about what would happen if it was raining, and so I did not tell a lie. So the statement \(P \to Q\) cannot be false in this case and so we consider it to be true.

Progress Check 1.4: xplorations with Conditional Statements

1 . Consider the following sentence:

If \(x\) is a positive real number, then \(x^2 + 8x\) is a positive real number.

Although the hypothesis and conclusion of this conditional sentence are not statements, the conditional sentence itself can be considered to be a statement as long as we know what possible numbers may be used for the variable \(x\). From the context of this sentence, it seems that we can substitute any positive real number for \(x\). We can also substitute 0 for \(x\) or a negative real number for x provided that we are willing to work with a false hypothesis in the conditional statement. (In Chapter 2 , we will learn how to be more careful and precise with these types of conditional statements.)

(a) Notice that if \(x = -3\), then \(x^2 + 8x = -15\), which is negative. Does this mean that the given conditional statement is false?

(b) Notice that if \(x = 4\), then \(x^2 + 8x = 48\), which is positive. Does this mean that the given conditional statement is true?

(c) Do you think this conditional statement is true or false? Record the results for at least five different examples where the hypothesis of this conditional statement is true.

2 . “If \(n\) is a positive integer, then \(n^2 - n +41\) is a prime number.” (Remember that a prime number is a positive integer greater than 1 whose only positive factors are 1 and itself.) To explore whether or not this statement is true, try using (and recording your results) for \(n = 1\), \(n = 2\), \(n = 3\), \(n = 4\), \(n = 5\), and \(n = 10\). Then record the results for at least four other values of \(n\). Does this conditional statement appear to be true?

Further Remarks about Conditional Statements

Suppose that Ed has exactly $52 in his wallet. The following four statements will use the four possible truth combinations for the hypothesis and conclusion of a conditional statement.

• If Ed has exactly $52 in his wallet, then he has $20 in his wallet. This is a true statement. Notice that both the hypothesis and the conclusion are true.
• If Ed has exactly $52 in his wallet, then he has $100 in his wallet. This statement is false. Notice that the hypothesis is true and the conclusion is false.
• If Ed has $100 in his wallet, then he has at least $50 in his wallet. This statement is true regardless of how much money he has in his wallet. In this case, the hypothesis is false and the conclusion is true.

This is admittedly a contrived example but it does illustrate that the conventions for the truth value of a conditional statement make sense. The message is that in order to be complete in mathematics, we need to have conventions about when a conditional statement is true and when it is false.

If \(n\) is a positive integer, then \((n^2 - n + 41)\) is a prime number.

Perhaps for all of the values you tried for \(n\), \((n^2 - n + 41)\) turned out to be a prime number. However, if we try \(n = 41\), we ge \(n^2 - n + 41 = 41^2 - 41 + 41\) \(n^2 - n + 41 = 41^2\) So in the case where \(n = 41\), the hypothesis is true (41 is a positive integer) and the conclusion is false \(41^2\) is not prime. Therefore, 41 is a counterexample for this conjecture and the conditional statement “If \(n\) is a positive integer, then \((n^2 - n + 41)\) is a prime number” is false. There are other counterexamples (such as \(n = 42\), \(n = 45\), and \(n = 50\)), but only one counterexample is needed to prove that the statement is false.

• Although one example can be used to prove that a conditional statement is false, in most cases, we cannot use examples to prove that a conditional statement is true. For example, in Progress Check 1.4 , we substituted values for \(x\) for the conditional statement “If \(x\) is a positive real number, then \(x^2 + 8x\) is a positive real number.” For every positive real number used for \(x\), we saw that \(x^2 + 8x\) was positive. However, this does not prove the conditional statement to be true because it is impossible to substitute every positive real number for \(x\). So, although we may believe this statement is true, to be able to conclude it is true, we need to write a mathematical proof. Methods of proof will be discussed in Section 1.2 and Chapter 3 .

Progress Check 1.5: Working with a Conditional Statement

The following statement is a true statement, which is proven in many calculus texts.

If the function \(f\) is differentiable at \(a\), then the function \(f\) is continuous at \(a\).

Using only this true statement, is it possible to make a conclusion about the function in each of the following cases?

• It is known that the function \(f\), where \(f(x) = \sin x\), is differentiable at 0.
• It is known that the function \(f\), where \(f(x) = \sqrt[3]x\), is not differentiable at 0.
• It is known that the function \(f\), where \(f(x) = |x|\), is continuous at 0.
• It is known that the function \(f\), where \(f(x) = \dfrac{|x|}{x}\) is not continuous at 0.

Closure Properties of Number Systems

The primary number system used in algebra and calculus is the real number system . We usually use the symbol R to stand for the set of all real numbers. The real numbers consist of the rational numbers and the irrational numbers. The rational numbers are those real numbers that can be written as a quotient of two integers (with a nonzero denominator), and the irrational numbers are those real numbers that cannot be written as a quotient of two integers. That is, a rational number can be written in the form of a fraction, and an irrational number cannot be written in the form of a fraction. Some common irrational numbers are \(\sqrt2\), \(\pi\) and \(e\). We usually use the symbol \(\mathbb{Q}\) to represent the set of all rational numbers. (The letter \(\mathbb{Q}\) is used because rational numbers are quotients of integers.) There is no standard symbol for the set of all irrational numbers.

Perhaps the most basic number system used in mathematics is the set of natural numbers . The natural numbers consist of the positive whole numbers such as 1, 2, 3, 107, and 203. We will use the symbol \(\mathbb{N}\) to stand for the set of natural numbers. Another basic number system that we will be working with is the set of integers . The integers consist of zero, the positive whole numbers, and the negatives of the positive whole numbers. If \(n\) is an integer, we can write \(n = \dfrac{n}{1}\). So each integer is a rational number and hence also a real number.

We will use the letter \(\mathbb{Z}\) to stand for the set of integers. (The letter \(\mathbb{Z}\) is from the German word, \(Zahlen\), for numbers.) Three of the basic properties of the integers are that the set \(\mathbb{Z}\) is closed under addition , the set \(\mathbb{Z}\) is closed under multiplication , and the set of integers is closed under subtraction. This means that

• If \(x\) and \(y\) are integers, then \(x + y\) is an integer;
• If \(x\) and \(y\) are integers, then \(x \cdot y\) is an integer; and
• If \(x\) and \(y\) are integers, then \(x - y\) is an integer.

Notice that these so-called closure properties are defined in terms of conditional statements. This means that if we can find one instance where the hypothesis is true and the conclusion is false, then the conditional statement is false.

Example 1.6: Closure

• In order for the set of natural numbers to be closed under subtraction, the following conditional statement would have to be true: If \(x\) and \(y\) are natural numbers, then \(x - y\) is a natural number. However, since 5 and 8 are natural numbers, \(5 - 8 = -3\), which is not a natural number, this conditional statement is false. Therefore, the set of natural numbers is not closed under subtraction.
• We can use the rules for multiplying fractions and the closure rules for the integers to show that the rational numbers are closed under multiplication. If \(\dfrac{a}{b}\) and \(\dfrac{c}{d}\) are rational numbers (so \(a\), \(b\), \(c\), and \(d\) are integers and \(b\) and \(d\) are not zero), then \(\dfrac{a}{b} \cdot \dfrac{c}{d} = \dfrac{ac}{bd}.\) Since the integers are closed under multiplication, we know that \(ac\) and \(bd\) are integers and since \(b \ne 0\) and \(d \ne 0\), \(bd \ne 0\). Hence, \(\dfrac{ac}{bd}\) is a rational number and this shows that the rational numbers are closed under multiplication.

Progress Check 1.7: Closure Properties

Answer each of the following questions.

• Is the set of rational numbers closed under addition? Explain.
• Is the set of integers closed under division? Explain.
• Is the set of rational numbers closed under subtraction? Explain.
• Which of the following sentences are statements? (a) \(3^2 + 4^2 = 5^2.\) (b) \(a^2 + b^2 = c^2.\) (c) There exists integers \(a\), \(b\), and \(c\) such that \(a^2 + b^2 = c^2.\) (d) If \(x^2 = 4\), then \(x = 2.\) (e) For each real number \(x\), if \(x^2 = 4\), then \(x = 2.\) (f) For each real number \(t\), \(\sin^2t + \cos^2t = 1.\) (g) \(\sin x < \sin (\frac{\pi}{4}).\) (h) If \(n\) is a prime number, then \(n^2\) has three positive factors. (i) 1 + \(\tan^2 \theta = \text{sec}^2 \theta.\) (j) Every rectangle is a parallelogram. (k) Every even natural number greater than or equal to 4 is the sum of two prime numbers.
• Identify the hypothesis and the conclusion for each of the following conditional statements. (a) If \(n\) is a prime number, then \(n^2\) has three positive factors. (b) If \(a\) is an irrational number and \(b\) is an irrational number, then \(a \cdot b\) is an irrational number. (c) If \(p\) is a prime number, then \(p = 2\) or \(p\) is an odd number. (d) If \(p\) is a prime number and \(p \ne 2\) or \(p\) is an odd number. (e) \(p \ne 2\) or \(p\) is a even number, then \(p\) is not prime.
• Determine whether each of the following conditional statements is true or false. (a) If 10 < 7, then 3 = 4. (b) If 7 < 10, then 3 = 4. (c) If 10 < 7, then 3 + 5 = 8. (d) If 7 < 10, then 3 + 5 = 8.
• Determine the conditions under which each of the following conditional sentences will be a true statement. (a) If a + 2 = 5, then 8 < 5. (b) If 5 < 8, then a + 2 = 5.
• Let \(P\) be the statement “Student X passed every assignment in Calculus I,” and let \(Q\) be the statement “Student X received a grade of C or better in Calculus I.” (a) What does it mean for \(P\) to be true? What does it mean for \(Q\) to be true? (b) Suppose that Student X passed every assignment in Calculus I and received a grade of B-, and that the instructor made the statement \(P \to Q\). Would you say that the instructor lied or told the truth? (c) Suppose that Student X passed every assignment in Calculus I and received a grade of C-, and that the instructor made the statement \(P \to Q\). Would you say that the instructor lied or told the truth? (d) Now suppose that Student X did not pass two assignments in Calculus I and received a grade of D, and that the instructor made the statement \(P \to Q\). Would you say that the instructor lied or told the truth? (e) How are Parts ( 5b ), ( 5c ), and ( 5d ) related to the truth table for \(P \to Q\)?

Theorem If f is a quadratic function of the form \(f(x) = ax^2 + bx + c\) and a < 0, then the function f has a maximum value when \(x = \dfrac{-b}{2a}\). Using only this theorem, what can be concluded about the functions given by the following formulas? (a) \(g (x) = -8x^2 + 5x - 2\) (b) \(h (x) = -\dfrac{1}{3}x^2 + 3x\) (c) \(k (x) = 8x^2 - 5x - 7\) (d) \(j (x) = -\dfrac{71}{99}x^2 +210\) (e) \(f (x) = -4x^2 - 3x + 7\) (f) \(F (x) = -x^4 + x^3 + 9\)

Theorem If \(f\) is a quadratic function of the form \(f(x) = ax^2 + bx + c\) and ac < 0, then the function \(f\) has two x-intercepts.

Using only this theorem, what can be concluded about the functions given by the following formulas? (a) \(g (x) = -8x^2 + 5x - 2\) (b) \(h (x) = -\dfrac{1}{3}x^2 + 3x\) (c) \(k (x) = 8x^2 - 5x - 7\) (d) \(j (x) = -\dfrac{71}{99}x^2 +210\) (e) \(f (x) = -4x^2 - 3x + 7\) (f) \(F (x) = -x^4 + x^3 + 9\)

Theorem A. If \(f\) is a cubic function of the form \(f (x) = x^3 - x + b\) and b > 1, then the function \(f\) has exactly one \(x\)-intercept. Following is another theorem about \(x\)-intercepts of functions: Theorem B . If \(f\) and \(g\) are functions with \(g (x) = k \cdot f (x)\), where \(k\) is a nonzero real number, then \(f\) and \(g\) have exactly the same \(x\)-intercepts.

Using only these two theorems and some simple algebraic manipulations, what can be concluded about the functions given by the following formulas? (a) \(f (x) = x^3 -x + 7\) (b) \(g (x) = x^3 + x +7\) (c) \(h (x) = -x^3 + x - 5\) (d) \(k (x) = 2x^3 + 2x + 3\) (e) \(r (x) = x^4 - x + 11\) (f) \(F (x) = 2x^3 - 2x + 7\)

• (a) Is the set of natural numbers closed under division? (b) Is the set of rational numbers closed under division? (c) Is the set of nonzero rational numbers closed under division? (d) Is the set of positive rational numbers closed under division? (e) Is the set of positive real numbers closed under subtraction? (f) Is the set of negative rational numbers closed under division? (g) Is the set of negative integers closed under addition? Explorations and Activities
• Exploring Propositions . In Progress Check 1.2 , we used exploration to show that certain statements were false and to make conjectures that certain statements were true. We can also use exploration to formulate a conjecture that we believe to be true. For example, if we calculate successive powers of \(2, (2^1, 2^2, 2^3, 2^4, 2^5, ...)\) and examine the units digits of these numbers, we could make the following conjectures (among others): \(\bullet\) If \(n\) is a natural number, then the units digit of \(2^n\) must be 2, 4, 6, or 8. \(\bullet\) The units digits of the successive powers of 2 repeat according to the pattern “2, 4, 8, 6.” (a) Is it possible to formulate a conjecture about the units digits of successive powers of \(4 (4^1, 4^2, 4^3, 4^4, 4^5,...)\)? If so, formulate at least one conjecture. (b) Is it possible to formulate a conjecture about the units digit of numbers of the form \(7^n - 2^n\), where \(n\) is a natural number? If so, formulate a conjecture in the form of a conditional statement in the form “If \(n\) is a natural number, then ... .” (c) Let \(f (x) = e^(2x)\). Determine the first eight derivatives of this function. What do you observe? Formulate a conjecture that appears to be true. The conjecture should be written as a conditional statement in the form, “If n is a natural number, then ... .”

Working with Conditional Statements and Negations

An error occurred trying to load this video.

Try refreshing the page, or contact customer support.

You must c C reate an account to continue watching

Register to access this and thousands of other videos.

As a member, you'll also get unlimited access to over 88,000 lessons in math, English, science, history, and more. Plus, get practice tests, quizzes, and personalized coaching to help you succeed.

Get unlimited access to over 88,000 lessons.

Already registered? Log in here for access

• 3:04 Working with…
• 4:37 Working with…

Alex has over five years of experience tutoring in math, science, and logic. They have a Master's in Physics from the University of Cincinnati, and both a B.S. in Applied Physics and a B.A. in Philosophy and Religion from UNC Pembroke.

Richard Cameron received his bachelor's degree in secondary mathematics education from Edinboro University. He has an active Pennsylvania teaching certificate and has taught high school math for three years teaching subjects from pre-algebra to calculus I.

Determining the Hypothesis, the Conclusion, and the Negation of a Conditional Statement

Determining the hypothesis and conclusion of a conditional statement.

Step 1: Consider the conditional statement, {eq}p \rightarrow q {/eq}, read as "if {eq}p {/eq} then {eq}q {/eq}." Note that sometimes {eq}\rightarrow {/eq} is written as {eq}\supset {/eq}.

Step 2: Identify the hypothesis and conclusion. The hypothesis appears on the left side of the logical connective ({eq}\rightarrow {/eq}) and the conclusion appears on the right side. If the conditional statement is written in plain English, the hypothesis appears after the word "if" and the conclusion appears after the word "then."

Working with Negations

Step 1: Identify the type of sentence. A sentence can be either closed, meaning it does not depend upon a variable or variables, or open, meaning it does depend upon a variable or variables.

Step 2: Identify the truth value of the sentence. A closed sentence is true or false (T or F); an open sentence is true or false (T or F) depending upon the value of the variable(s). If the sentence is open, you will not be able to determine the truth of it without particular values for the variables.

Step 3: Construct the negation of the sentence. The negation of sentence {eq}p {/eq} is written as {eq}\sim p {/eq} or {eq}\neg p {/eq} and has the opposite truth value of the original sentence. A double negation {eq}\sim \sim p {/eq} is equal to the original statement {eq}p {/eq}!

Determining the Hypothesis, the Conclusion, and the Negation of a Conditional Statement Vocabulary

Conditional statement : a statement of the form "if {eq}p {/eq} then {eq}q {/eq}," symbolically written as {eq}p \rightarrow q {/eq}, where {eq}\rightarrow {/eq} is the conditional statement logical connective.

Hypothesis : The first statement in a conditional statement, occurring after the word "if" or on the left side of the logical connective {eq}\rightarrow {/eq}.

Conclusion : The second statement in a conditional statement, said to follow from the hypothesis. It occurs after the word "then" or on the right side of the logical connective {eq}\rightarrow {/eq}.

Negation : Symbolized as {eq}\sim {/eq} or {eq}\neg {/eq}, and often read as "negation" or "not." The negation of a statement has the opposite truth value of the original statement; i.e., if {eq}p {/eq} is true, {eq}\sim p {/eq} is false.

Truth value : The truth or falsehood of a sentence, represented by T or F, respectively.

Closed sentence : A sentence whose truth value does not depend upon any variables, such as in "Michigan is part of the USA." The negation of this sentence would be "Michigan is not part of the USA."

Open sentence : A sentence whose truth value does depend upon a variable or variables, such as "She will do her homework" or {eq}x < 7 {/eq}, where the variables are "she" and {eq}x {/eq}, respectively. The negations of these sentences would be "She will not do her homework" and {eq}x \geq 7 {/eq}. Note that even without being able to assign truth values to the open sentences, we could still form their negations.

Let's take a look at three example problems to see this information in action. The first example will have you distinguish the hypothesis of a conditional statement, and the second will have you pick out the conclusion of a conditional statement. Example 3 will have you determine the type of sentence (closed or open) of mathematical statements and then form their negations.

Determining the Hypothesis, the Conclusion, and the Negation of a Conditional Statement: Example 1

Let {eq}p {/eq} be "All cats are mammals," and let {eq}q {/eq} be "My cat is a mammal." The conditional statement {eq}p \rightarrow q {/eq} reads as "If all cats are mammals, then my cat is a mammal." What is the hypothesis of this statement?

(a) My cat is a mammal

(b) Every cat is a mammal, so mine is too

(c) All cats are mammals

(d) Mammals are cats

The hypothesis is "All cats are mammals" (c) because it occurs on the left side of the arrow, or after the word "if"; the conclusion is "My cat is a mammal" because it occurs on the right side of the arrow, or after the word "then."

Determining the Hypothesis, the Conclusion, and the Negation of a Conditional Statement: Example 2

Consider the sentence, "If it rains today, tomorrow will be sunny." What is the conclusion of this conditional statement?

(a) Tomorrow will be sunny

(b) It's raining today

(c) If it's rainy today, tomorrow will always be sunny

(d) Because it rained today, tomorrow will be sunny

The conclusion of the statement always follows the word "then." Thus we see that the conclusion is "tomorrow will be sunny." The corresponding choice is answer (a).

Determining the Hypothesis, the Conclusion, and the Negation of a Conditional Statement: Example 3

Identify whether each sentence is closed or open, and then construct its negation:

1. {eq}11 + 3 = 20 {/eq}

2. {eq}x^2 - 5 \leq y {/eq}

Sentence 1 is closed; its truth value does not depend upon any variables. It will always be false, which we find upon addition of 11 and 3. Its negation, therefore, will be true. To construct its negation, we change the {eq}= {/eq} to {eq}\neq {/eq}. The negation of a statement involving {eq}= {/eq} will always have {eq}\neq {/eq}, and vice versa!

Sentence 2 is open; its truth depends upon two variables, both {eq}x {/eq} and {eq}y {/eq}. It will sometimes be true and sometimes false, for different values of {eq}x {/eq} and {eq}y {/eq}; regardless, its negation will always have the opposite truth value. The sentence with the opposite truth value would be {eq}x^2 - 5 > y {/eq}, and thus it is the original sentence's negation. Note how we flipped the direction of the inequality and removed the "equal to" component. If we were to negate the negation, we would have to flip the sign again and add the "equal to" component back in. Follow this general rule when negating inequalities!

Recently updated on Study.com

• Teacher Resources

Understanding the Inverse Statement | Explained with Examples and Significance

Inverse statement, the inverse statement is a logical statement that is formed by negating both the hypothesis and the conclusion of an original conditional statement.

The inverse statement is a logical statement that is formed by negating both the hypothesis and the conclusion of an original conditional statement.

In general, a conditional statement consists of two parts: the hypothesis and the conclusion. The hypothesis is the “if” part of the statement, and the conclusion is the “then” part. For example, consider the conditional statement:

“If it is raining, then the ground is wet.”

To form the inverse statement, we simply negate both the hypothesis and the conclusion of the original statement. In this case, the inverse statement would be:

“If it is not raining, then the ground is not wet.”

The key idea in the inverse statement is that if the original statement is true, then the inverse statement may or may not be true. The truth value of the inverse is independent of the original statement. So, if the original statement is true, the inverse statement can be either true or false, and if the original statement is false, the inverse can also be either true or false.

It’s important to note that the inverse statement is not always logically equivalent to the original conditional statement. In other words, the truth value of the inverse statement does not always match the truth value of the original statement. However, if both the original statement and its inverse are true or both are false, then they are logically equivalent.

More Answers:

Recent posts, ramses ii a prominent pharaoh and legacy of ancient egypt.

Ramses II (c. 1279–1213 BCE) Ramses II, also known as Ramses the Great, was one of the most prominent and powerful pharaohs of ancient Egypt.

Formula for cyclic adenosine monophosphate & Its Significance

Is the formula of cyclic adenosine monophosphate (cAMP) $ce{C_{10}H_{11}N_{5}O_{6}P}$ or $ce{C_{10}H_{12}N_{5}O_{6}P}$? Does it matter? The correct formula for cyclic adenosine monophosphate (cAMP) is $ce{C_{10}H_{11}N_{5}O_{6}P}$. The

Development of a Turtle Inside its Egg

How does a turtle develop inside its egg? The development of a turtle inside its egg is a fascinating process that involves several stages and

The Essential Molecule in Photosynthesis for Energy and Biomass

Why does photosynthesis specifically produce glucose? Photosynthesis is the biological process by which plants, algae, and some bacteria convert sunlight, carbon dioxide (CO2), and water

How the Human Body Recycles its Energy Currency

Source for “The human body recycles its body weight of ATP each day”? The statement that “the human body recycles its body weight of ATP

Don't Miss Out! Sign Up Now!

Sign up now to get started for free!

Negating both the and of a . For example, the inverse of "If it is raining then the grass is wet" is "If it is not raining then the grass is not wet". Note: As in the example, a proposition may be true but its inverse may be false.     , ,   Copyright © 2000 by Bruce Simmons All rights reserved 2.5 Equivalent Statements Learning objectives. After completing this section, you should be able to: Determine whether two statements are logically equivalent using a truth table. Compose the converse, inverse, and contrapositive of a conditional statement Have you ever had a conversation with or sent a note to someone, only to have them misunderstand what you intended to convey? The way you choose to express your ideas can be as, or even more, important than what you are saying. If your goal is to convince someone that what you are saying is correct, you will not want to alienate them by choosing your words poorly. Logical arguments can be stated in many different ways that still ultimately result in the same valid conclusion. Part of the art of constructing a persuasive argument is knowing how to arrange the facts and conclusion to elicit the desired response from the intended audience. In this section, you will learn how to determine whether two statements are logically equivalent using truth tables, and then you will apply this knowledge to compose logically equivalent forms of the conditional statement. Developing this skill will provide the additional skills and knowledge needed to construct well-reasoned, persuasive arguments that can be customized to address specific audiences. An alternate way to think about logical equivalence is that the truth values have to match. That is, whenever p p is true, q q is also true, and whenever p p is false, q q is also false. Determine Logical Equivalence Two statements, p p and q q , are logically equivalent when p ↔ q p ↔ q is a valid argument, or when the last column of the truth table consists of only true values. When a logical statement is always true, it is known as a tautology . To determine whether two statements p p and q q are logically equivalent, construct a truth table for p ↔ q p ↔ q and determine whether it valid. If the last column is all true, the argument is a tautology, it is valid, and p p is logically equivalent to q q ; otherwise, p p is not logically equivalent to q q . Example 2.22 Determining logical equivalence with a truth table. Create a truth table to determine whether the following compound statements are logically equivalent. p → q ; p → q ; ~ p → ~ q ~ p → ~ q p → q ; p → q ; ~ p ∨ q ~ p ∨ q T T T F F T T F F F T T F T T T F F F F T T T T Because the last column it not all true, the biconditional is not valid and the statement p → q p → q is not logically equivalent to the statement ~ p → ~ q ~ p → ~ q . T T T F T T F F F F F T T T T F F T T T Because the last column is true for every entry, the biconditional is valid and the statement p → q p → q is logically equivalent to the statement ~ p ∨ q ~ p ∨ q . Symbolically, p → q ≡ ~ p ∨ q . p → q ≡ ~ p ∨ q . Your Turn 2.22 Compose the converse, inverse, and contrapositive of a conditional statement. The converse , inverse , and contrapositive are variations of the conditional statement, p → q . p → q . The converse is if q q then p p , and it is formed by interchanging the hypothesis and the conclusion. The converse is logically equivalent to the inverse. The inverse is if ~ p ~ p then ~ q ~ q , and it is formed by negating both the hypothesis and the conclusion. The inverse is logically equivalent to the converse. The contrapositive is if ~ q ~ q then ~ p ~ p , and it is formed by interchanging and negating both the hypothesis and the conclusion. The contrapositive is logically equivalent to the conditional. The table below shows how these variations are presented symbolically. Conditional Contrapositive Converse Inverse T T F F T T T T T F F T F F T T F T T F T T F F F F T T T T T T Example 2.23 Writing the converse, inverse, and contrapositive of a conditional statement. Use the statements, p p : Harry is a wizard and q q : Hermione is a witch, to write the following statements: Write the conditional statement, p → q p → q , in words. Write the converse statement, q → p q → p , in words. Write the inverse statement, ~ p → ~ q ~ p → ~ q , in words. Write the contrapositive statement, ~ q → ~ p ~ q → ~ p , in words. The conditional statement takes the form, “if p p , then q q ,” so the conditional statement is: “If Harry is a wizard, then Hermione is a witch.” Remember the if … then … words are the connectives that form the conditional statement. The converse swaps or interchanges the hypothesis, p p , with the conclusion, q q . It has the form, “if q q , then p p .” So, the converse is: "If Hermione is a witch, then Harry is a wizard." To construct the inverse of a statement, negate both the hypothesis and the conclusion. The inverse has the form, “if ~ p ~ p , then ~ q ~ q ,” so the inverse is: "If Harry is not a wizard, then Hermione is not a witch." The contrapositive is formed by negating and interchanging both the hypothesis and conclusion. It has the form, “if ~ q ~ q , then ~ p ~ p ," so the contrapositive statement is: "If Hermione is not a witch, then Harry is not a wizard." Your Turn 2.23 Example 2.24, identifying the converse, inverse, and contrapositive. Use the conditional statement, “If all dogs bark, then Lassie likes to bark,” to identify the following. Write the hypothesis of the conditional statement and label it with a p p . Write the conclusion of the conditional statement and label it with a q q . Identify the following statement as the converse, inverse, or contrapositive: “If Lassie likes to bark, then all dogs bark.” Identify the following statement as the converse, inverse, or contrapositive: “If Lassie does not like to bark, then some dogs do not bark.” Which statement is logically equivalent to the conditional statement? The hypothesis is the phrase following the if . The answer is p p : All dogs bark. Notice, the word if is not included as part of the hypothesis. The conclusion of a conditional statement is the phrase following the then . The word then is not included when stating the conclusion. The answer is: q q : Lassie likes to bark. “Lassie likes to bark” is q q and “All dogs bark” is p p . So, “If Lassie likes to bark, then all dogs bark,” has the form “if q q , then p p ,” which is the form of the converse. “Lassie does not like to bark” is ~ q ~ q and “Some dogs do not bark” is ~ p ~ p . The statement, “If Lassie does not like to bark, then some dogs do not bark,” has the form “if ~ q ~ q , then ~ p ~ p ,” which is the form of the contrapositive. The contrapositive ~ q → ~ p ~ q → ~ p is logically equivalent to the conditional statement p → q . p → q . Your Turn 2.24 Example 2.25, determining the truth value of the converse, inverse, and contrapositive. Assume the conditional statement, p → q : p → q : “If Chadwick Boseman was an actor, then Chadwick Boseman did not star in the movie Black Panther ” is false, and use it to answer the following questions. Write the converse of the statement in words and determine its truth value. Write the inverse of the statement in words and determine its truth value. Write the contrapositive of the statement in words and determine its truth value. The only way the conditional statement can be false is if the hypothesis, p p : Chadwick Boseman was an actor, is true and the conclusion, q q : Chadwick Boseman did not star in the movie Black Panther , is false. The converse is q → p , q → p , and it is written in words as: “If Chadwick Boseman did not star in the movie Black Panther , then Chadwick Boseman was an actor.” This statement is true, because false → → true is true. The inverse has the form “ ~ p → ~ q . ~ p → ~ q . ” The written form is: “If Chadwick Boseman was not an actor, then Chadwick Boseman starred in the movie Black Panther .” Because p p is true, and q q is false, ~ p ~ p is false, and ~ q ~ q is true. This means the inverse is false → → true, which is true. Alternatively, from Question 1, the converse is true, and because the inverse is logically equivalent to the converse it must also be true. The contrapositive is logically equivalent to the conditional. Because the conditional is false, the contrapositive is also false. This can be confirmed by looking at the truth values of the contrapositive statement. The contrapositive has the form “ ~ q → ~ p ~ q → ~ p .” Because q q is false and p p is true, ~ q ~ q is true and ~ p ~ p is false. Therefore, ~ q → ~ p ~ q → ~ p is true → → false, which is false. The written form of the contrapositive is “If Chadwick Boseman starred in the movie Black Panther , then Chadwick Boseman was not an actor.” Your Turn 2.25 Check your understanding, section 2.5 exercises. Write the conditional statement p → q in words. Write the converse statement q → p in words. Write the inverse statement ~ p → ~ q in words. Write the contrapositive statement ~ q → ~ p in words. As an Amazon Associate we earn from qualifying purchases. This book may not be used in the training of large language models or otherwise be ingested into large language models or generative AI offerings without OpenStax's permission. Want to cite, share, or modify this book? This book uses the Creative Commons Attribution License and you must attribute OpenStax. Access for free at https://openstax.org/books/contemporary-mathematics/pages/1-introduction Authors: Donna Kirk Publisher/website: OpenStax Book title: Contemporary Mathematics Publication date: Mar 22, 2023 Location: Houston, Texas Book URL: https://openstax.org/books/contemporary-mathematics/pages/1-introduction Section URL: https://openstax.org/books/contemporary-mathematics/pages/2-5-equivalent-statements © Dec 21, 2023 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License . The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University. Get in touch with us Are you sure you want to logout? Please select your grade. Earth and space Conditional Statement Key concepts. Write definitions as conditional statements Verify statements Write the converse, inverse and the contrapositive of the conditional statement Write definitions as biconditional statements Definition  A conditional statement is a logical statement that has two parts, a hypothesis and a conclusion. When a conditional statement is written in if-then form , the “if” part contains the hypothesis, and the “then” part contains the conclusion .  Here is an example:  Re-write a Statement in if-then form  Example 1 : Re-write the conditional statement in if-then form.   All birds have feathers.   Two angles are supplementary if they are a linear pair.  First, identify the hypothesis and the conclusion . When you rewrite the statement in if-then form, you may need to reword the hypothesis or conclusion.  1. All birds have feathers .  If an animal is a bird, then it has feathers.   2. Two angles are supplementary if they are a linear pair.   If two angles are a linear pair, then they are supplementary.   Negation and Verifying Statements  The negation of a statement is the opposite of the original statement. Notice that Statement 2 is already negative, so its negation is positive.  Statement 1: The ball is red. Statement 2: The cat is not black.  Statement 2: The cat is not black. Negation 2: The cat is black.   Verifying Statements  Conditional statements can be true or false. To show that a conditional statement is true, you must prove that the conclusion is true every time the hypothesis is true. To show that a conditional statement is false, you need to give only one counterexample.  Related Conditionals  To write the converse of a conditional statement, exchange the hypothesis and conclusion .   To write the inverse of a conditional statement, negate both the hypothesis and the conclusion. To write the contrapositive , first, write the converse and then negate both the hypothesis and the conclusion.  Write four related conditional statements  Example 2:    Write the if-then form, the converse, the inverse, and the contrapositive of the conditional statement “Guitar players are musicians.” Decide whether each statement is true or false .  Solution:   If-then form: If you are a guitar player, then you are a musician.  True, guitars players are musicians.  Converse: If you are a musician, then you are a guitar player.  False, not all musicians play the guitar.  Inverse: If you are not guitar player, then you are not a musician.  False, even if you don’t play a guitar, you can still be a musician.  Contrapositive: If you are not a musician, then you are not a guitar player.  True, a person who is not a musician cannot be a guitar player.  Equivalent Statements   A conditional statement and its contrapositive are either both true or both false. Similarly, the converse and inverse of a conditional statement are either both true or both false. Pairs of statements such as these are called equivalent statements . In general, when two statements are both true or both false, they are called equivalent statements.  You can write a definition as a conditional statement in if-then form or as its converse. Both the conditional statement and its converse are true. For example, consider the definition of perpendicular lines .  Perpendicular Lines If two lines intersect to form a right angle, then they are perpendicular lines .  The definition can also be written using the converse:   If two lines are perpendicular lines, then they intersect to form a right angle.  You can write “line l is perpendicular to line m” as l ⊥ m.  Use Definitions  Decide whether each statement about the diagram is true. Explain your answer using the definitions you have learned.  −𝐀𝐂 ↔ ⊥ 𝐁𝐃 ↔  ∠𝐀𝐄𝐁 and ∠𝐂𝐄𝐁 are a linear pair.  →−𝐄𝐀→ and →−𝐄𝐁→ are opposite rays.  This statement is true. The right angle symbol in the diagram indicates that the lines intersect to form a right angle. So you can say the lines are perpendicular.  This statement is true. By definition, if the noncommon sides of adjacent angles are opposite rays, then the angles are opposite rays, then the angles are a linear pair. Because 𝐄𝐀 → and →−𝐄𝐂→ are opposite rays, ∠𝐀𝐄𝐁 and ∠𝐂𝐄𝐁 are a linear pair.    This statement is false. Point E does not lie on the same line as A and B, so the rays are not opposite rays.  Bi-conditional Statements  Definition  . When a conditional statement and its converse are both true, you can write them as a single biconditional statement . A biconditional statement is a statement that contains the phrase “if and only if.”   Any valid definition can be written as a biconditional statement.  Write a Bi-conditional  Example 4:    Write the definition of perpendicular lines as a biconditional.  Definition: If two lines intersect to form a right angle, then they are perpendicular.   Converse: If two lines perpendicular, then they intersect to form a right angle.   Biconditional: Two lines are perpendicular, if and only if they intersect to form a right angle.   Question 1 :  Rewrite the conditional statement in if-then form.  Only people who are registered are allowed to vote.   If one is allowed to vote, then one is registered.   Question 2 : Write the if-then form, the converse, the inverse, and the contrapositive of the following statement.  3x + 10 = 16, because x = 2.   If-then form:   If 3x + 10 = 16, then x = 2.   If x = 2, then 3x + 10 = 16.   If 3x + 10 is not equal to 16, then x is not equal to 2.   Contrapositive:  If x is not equal to 2, then 3x + 10 is not equal to 16.   Question 3 : Decide whether the statement is true or false. If false, provide a counterexample.  If a polygon has five sides, then it is a regular pentagon.   False statement  Counterexample: If a polygon has five sides of unequal length, then it is not a regular pentagon. The sides of a regular pentagon should be equal in length.   Question 4 : Rewrite the definition as a biconditional statement.  An angle with a measure between 90 degrees and 180 degrees is called obtuse.   An angle is called obtuse if and only if it measures between 90 degrees and 180 degrees.   Key Concepts Covered   To write a conditional statement in if-then form, find the hypothesis and then the conclusion.   Converse: To write the converse of a conditional statement, exchange the hypothesis and conclusion.  Inverse: To write the inverse of a conditional statement, negate both the hypothesis and the conclusion.  Contrapositive: To write the contrapositive, first write the converse and then negate both the hypothesis and the conclusion.   When two statements are both true or both false, they are called equivalent statements .  A biconditional statement is a statement that contains the phrase “if and only if.”  In triangle ABC, AD is a median. If the area of ΔABD is 15 cm sq. then find the area of ΔABC. ABCD is a parallelogram and BPC is a triangle with P falling on AD. If the area of parallelogram ABCD= 26 cm 2 , find the area of triangle BPC. PQRS is a parallelogram and PQT is a triangle with T falling on RS. If area of triangle PQT = 18 cm 2 , then find the area of parallelogram PQRS. ABCD is a parallelogram where E is a point on AD. Area of ΔBCE = 21 cm 2 . If CD = 6 cm, then find the length of AF. The area of triangle ABC is 15 cm sq. If ΔABC and a parallelogram ABPD are on the same base and between the same parallel lines then what is the area of parallelogram ABPD. The area of parallelogram PQRS is 88 cm sq. A perpendicular from S is drawn to intersect PQ at M. If SM = 8 cm, then find the length of PQ. Amy needs to order a shade for a triangular-shaped window that has a base of 6 feet and a height of 4 feet. What is the area of the shade? Monica has a triangular piece of fabric. The height of the triangle is 15 inches and the triangle’s base is 6 inches. Monica says that the area of the fabric is 90 square inches. What error did Monica make? Explain your answer. The sixth-grade art students are making a mosaic using tiles in the shape of right triangle. The two sides that meet to form a right angle are 3 centimeters and 5 centimeters long. If there are 200 tiles in the mosaic, what is the area of the mosaic? A parallelogram with area 301 has a base of 35. What is its height? Related topics Addition and Multiplication Using Counters & Bar-Diagrams Introduction: We can find the solution to the word problem by solving it. Here, in this topic, we can use 3 methods to find the solution. 1. Add using counters 2. Use factors to get the product 3. Write equations to find the unknown. Addition Equation: 8+8+8 =? Multiplication equation: 3×8=? Example 1: Andrew has […] Dilation: Definitions, Characteristics, and Similarities Understanding Dilation A dilation is a transformation that produces an image that is of the same shape and different sizes. Dilation that creates a larger image is called enlargement. Describing Dilation Dilation of Scale Factor 2 The following figure undergoes a dilation with a scale factor of 2 giving an image A’ (2, 4), B’ […] How to Write and Interpret Numerical Expressions? Write numerical expressions What is the Meaning of Numerical Expression? A numerical expression is a combination of numbers and integers using basic operations such as addition, subtraction, multiplication, or division. The word PEMDAS stands for: P → Parentheses E → Exponents M → Multiplication D → Division  A → Addition S → Subtraction         Some examples […] System of Linear Inequalities and Equations Introduction: Systems of Linear Inequalities: A system of linear inequalities is a set of two or more linear inequalities in the same variables. The following example illustrates this, y < x + 2…………..Inequality 1 y ≥ 2x − 1…………Inequality 2 Solution of a System of Linear Inequalities: A solution of a system of linear inequalities […] Other topics How to Find the Area of Rectangle? How to Solve Right Triangles? Ways to Simplify Algebraic Expressions Chapter 7 Logic 7.5 Equivalent Statements Learning Objectives By the end of this section, you will be able to: Determine whether two statements are logically equivalent using a truth table. Compose the converse, inverse, and contrapositive of a conditional statement. Have you ever had a conversation with or sent a note to someone, only to have them misunderstand what you intended to convey? The way you choose to express your ideas can be as, or even more, important than what you are saying. If your goal is to convince someone that what you are saying is correct, you will not want to alienate them by choosing your words poorly. Logical arguments can be stated in many different ways that still ultimately result in the same valid conclusion. Part of the art of constructing a persuasive argument is knowing how to arrange the facts and conclusion to elicit the desired response from the intended audience. In this section, you will learn how to determine whether two statements are  logically equivalent  using truth tables, and then you will apply this knowledge to compose logically equivalent forms of the conditional statement. Developing this skill will provide the additional skills and knowledge needed to construct well-reasoned, persuasive arguments that can be customized to address specific audiences. Determine Logical Equivalence Two statements, [latex]p[/latex] and [latex]q[/latex], are logically equivalent when [latex]p \leftrightarrow  q[/latex] is a valid argument, or when the last column of the truth table consists of only true values. When a logical statement is always true, it is known as a tautology . To determine whether two statements [latex]p[/latex]  and [latex]q[/latex]  are logically equivalent, construct a truth table for [latex]p \leftrightarrow q[/latex]  and determine whether it valid. If the last column is all true, the argument is a tautology, it is valid, and [latex]p[/latex] is logically equivalent to [latex]q[/latex] ; otherwise, [latex]p[/latex] is not logically equivalent to [latex]q[/latex]. q q "> An alternate way to think about logical equivalence is that the truth values have to match. That is, whenever [latex]p[/latex]  is true, [latex]q[/latex]  is also true, and whenever [latex]p[/latex]  is false, [latex]q[/latex]  is also false. Create a truth table to determine whether the following compound statements are logically equivalent. a) [latex]p\rightarrow q; \sim{p} \rightarrow \sim {q}[/latex] Construct a truth table for the biconditional formed by using the first statement as the hypothesis and the second statement as the conclusion, [latex](p \rightarrow q)\leftrightarrow(\sim p \rightarrow \sim q)[/latex] . [latex]\begin{array}{|c|c|c|c|c|c|c|c|} \hline p&q&p\rightarrow q& \sim{p}&\sim{q}&\sim{p} \rightarrow \sim{q} &(p \rightarrow q) \leftrightarrow (\sim{p} \rightarrow \sim{q})\\ \hline \text{T}&\text{T}&\text{T}&\text{F}&\text{F}&\text{T}&\textbf{T} \\ \hline \text{T}&\text{F}&\text{F}&\text{F}&\text{T}&\text{T}&\textbf{F} \\ \hline \text{F}&\text{T}&\text{T}&\text{T}&\text{F}&\text{F}&\textbf{F} \\ \hline \text{F}&\text{F}&\text{T}&\text{T}&\text{T}&\text{T}&\textbf{T} \\ \hline \end{array}[/latex] Because the last column is not all true, the biconditional is not valid and the statement [latex]p \rightarrow q[/latex]  is not logically equivalent to the statement [latex]\sim{p} \rightarrow \sim {q}[/latex]. b) [latex]p \rightarrow q; \sim p \vee q[/latex] Construct a truth table for the biconditional formed by using the first statement as the hypothesis and the second statement as the conclusion, [latex](p \rightarrow q) \leftrightarrow (\sim{p} \vee q)[/latex]. [latex]\begin{array}{|c|c|c|c|c|c|c|} \hline p&q&p\rightarrow q& \sim{p}&\sim{p} \vee q &(p \rightarrow q) \leftrightarrow (\sim{p} \vee q)\\ \hline \text{T}&\text{T}&\text{T}&\text{F}&\text{T}&\textbf{T} \\ \hline \text{T}&\text{F}&\text{F}&\text{F}&\text{F}&\textbf{T} \\ \hline \text{F}&\text{T}&\text{T}&\text{T}&\text{T}&\textbf{T} \\ \hline \text{F}&\text{F}&\text{T}&\text{T}&\text{T}&\textbf{T} \\ \hline \end{array}[/latex] Because the last column is all true, the biconditional is valid and the statement [latex]p \rightarrow q[/latex]  is logically equivalent to the statement [latex]\sim{p} \vee q[/latex]. Symbollically, [latex]p \rightarrow q \equiv \sim{p} \vee q[/latex]. a) [latex]p\rightarrow q; \sim{q} \rightarrow \sim {p}[/latex] b) [latex]p \rightarrow q; p \vee \sim{q}[/latex] a) [latex]p \rightarrow q[/latex]  is logically equivalent  [latex]\sim{q} \rightarrow \sim {p}[/latex]. [latex]\begin{array}{|c|c|c|c|c|c|c|c|} \hline p&q&p\rightarrow q& \sim{q}&\sim{p}&\sim{q} \rightarrow \sim{p} &(p \rightarrow q) \leftrightarrow (\sim{q} \rightarrow \sim{p})\\ \hline \text{T}&\text{T}&\text{T}&\text{F}&\text{F}&\text{T}&\textbf{T} \\ \hline \text{T}&\text{F}&\text{F}&\text{T}&\text{F}&\text{F}&\textbf{T} \\ \hline \text{F}&\text{T}&\text{T}&\text{F}&\text{T}&\text{T}&\textbf{T} \\ \hline \text{F}&\text{F}&\text{T}&\text{T}&\text{T}&\text{T}&\textbf{T} \\ \hline \end{array}[/latex] b) [latex]p \rightarrow q[/latex]  is not logically equivalent [latex]p \vee \sim{q}[/latex].  [latex]\begin{array}{|c|c|c|c|c|c|c|} \hline p&q&p\rightarrow q& \sim{q}&p\vee \sim{q} &(p \rightarrow q) \leftrightarrow (p \vee \sim{q})\\ \hline \text{T}&\text{T}&\text{T}&\text{F}&\text{T}&\textbf{T} \\ \hline \text{T}&\text{F}&\text{F}&\text{T}&\text{T}&\textbf{F} \\ \hline \text{F}&\text{T}&\text{T}&\text{F}&\text{F}&\textbf{F} \\ \hline \text{F}&\text{F}&\text{T}&\text{T}&\text{T}&\textbf{T} \\ \hline \end{array}[/latex] Compose the Converse, Inverse, and Contrapositive of a Conditional Statement The  converse ,  inverse , and  contrapositive are variations of the conditional statement, [latex]p \rightarrow q[/latex]. The converse is if [latex]q[/latex] then [latex]p[/latex], and it is formed by interchanging the hypothesis and the conclusion. The converse is logically equivalent to the inverse. The inverse is if [latex]\sim{p}[/latex] then [latex]\sim{q}[/latex], and it is formed by negating both the hypothesis and the conclusion. The inverse is logically equivalent to the converse. The contrapositive is if [latex]\sim{q}[/latex] then [latex]\sim{p}[/latex], and it is formed by interchanging and negating both the hypothesis and the conclusion. The contrapositive is logically equivalent to the conditional. The table below shows how these variations are presented symbolically. [latex]\begin{array}{|c|c|c|c|c|c|c|c|c|} \hline & & & & \text{Conditional}&\text{Contrapositive} &\text{Converse}&\text{Inverse}\\ \hline p&q&\sim{p} & \sim{q} & p\rightarrow q& \sim{q}\rightarrow \sim{p}&q \rightarrow p &\sim{p} \rightarrow \sim{q}\\ \hline \text{T}&\text{T}&\text{F}&\text{F}&\text{T}&\text{T}&\text{T}&\text{T} \\ \hline \text{T}&\text{F}&\text{F}&\text{T}&\text{F}&\text{F}&\text{T}&\text{T} \\ \hline \text{F}&\text{T}&\text{T}&\text{F}&\text{T}&\text{T}&\text{F}&\text{F} \\ \hline \text{F}&\text{F}&\text{T}&\text{T}&\text{T}&\text{T}&\text{T}&\text{T} \\ \hline \end{array}[/latex] Use the statements [latex]p[/latex]: Harry is a wizard and [latex]q[/latex]: Hermione is a witch to write the following statements: a) Write the conditional statement, [latex]p \rightarrow q[/latex], in words. The conditional statement takes the form, “if [latex]p[/latex], then [latex]q[/latex],” so the conditional statement is: “If Harry is a wizard, then Hermione is a witch.” Remember the  if  …  then  … words are the connectives that form the conditional statement. b) Write the converse statement, [latex]q \rightarrow p[/latex], in words. The converse swaps or interchanges the hypothesis, [latex]p[/latex], with the conclusion, [latex]q[/latex]. It has the form, “if  q q "> [latex]q[/latex] , then  p p "> [latex]p[/latex] .” So, the converse is: “If Hermione is a witch, then Harry is a wizard.” c) Write the inverse statement, [latex]\sim{p} \rightarrow \sim{q}[/latex], in words. To construct the inverse of a statement, negate both the hypothesis and the conclusion. The inverse has the form, “if [latex]\sim{p}[/latex], then [latex]\sim{q}[/latex]” so the inverse is: “If Harry is not a wizard, then Hermione is not a witch.” d) Write the contrapositive statement, [latex]\sim{q} \rightarrow \sim{p}[/latex], in words. The contrapositive is formed by negating and interchanging both the hypothesis and conclusion. It has the form, “if [latex]\sim{q}[/latex], then,[latex]\sim{p}[/latex]” so the contrapositive statement is: “If Hermione is not a witch, then Harry is not a wizard.” Use the statements [latex]p[/latex]: Elvis Presly wore capes and [latex]q[/latex]: Some superheroes wear capes to write the following statements: a) If Elvis Presley wore capes, then some superheroes wear capes. b) If some superheroes wear capes, then Elvis Presley wore capes. c) If Elvis Presley did not wear capes, then no superheroes wear capes. d) If no superheroes wear capes, then Elvis Presley did not wear capes. Use the conditional statement, “If all dogs bark, then Lassie likes to bark,” to identify the following. a) Write the hypothesis of the conditional statement and label it with a [latex]p[/latex]. The hypothesis is the phrase following the  if . The answer is [latex]p[/latex]: All dogs bark. Notice, the word  if  is not included as part of the hypothesis. b) Write the conclusion of the conditional statement and label it with a [latex]q[/latex]. The conclusion of a conditional statement is the phrase following the  then . The word  then  is not included when stating the conclusion. The answer is: [latex]q[/latex]: Lassie likes to bark. c) Identify the following statement as the converse, inverse, or contrapositive: “If Lassie likes to bark, then all dogs bark.” “Lassie likes to bark” is [latex]q[/latex] and “All dogs bark” is [latex]p[/latex]. So, “If Lassie likes to bark, then all dogs bark,” has the form “if [latex]q[/latex], then [latex]p[/latex],” which is the form of the converse. d) Identify the following statement as the converse, inverse, or contrapositive: “If Lassie does not like to bark, then some dogs do not bark.” “Lassie does not like to bark” is [latex]\sim{q}[/latex] ~ q ~ q "> ~ q ~ q "> ~ q ~ q "> ~ q ~ q "> ~ q ~ q "> ~ q ~ q "> ~ q ~ q "> ~ q ~ q "> ~ q ~ q "> ~ q ~ q "> ~ q ~ q ">   and “Some dogs do not bark” is [latex]\sim{p}[/latex]. The statement, “If Lassie does not like to bark, then some dogs do not bark,” has the form “if [latex]\sim{q}[/latex], then [latex]\sim{p}[/latex],” which is the form of the contrapositive. e) Which statement is logically equivalent to the conditional statement? The contrapositive [latex]\sim{q} \rightarrow \sim{p}[/latex]  is logically equivalent to the conditional statement [latex]p \rightarrow q[/latex] p → q . p → q . "> . Use the conditional statement, “If Dora is an explorer, then Boots is a monkey,” to identify the following: c) Identify the following statement as the converse, inverse, or contrapositive: “If Dora is not an explorer, then Boots is not a monkey.” d) Identify the following statement as the converse, inverse, or contrapositive: “If Boots is a monkey, then Dora is an explorer.” e) Which statement is logically equivalent to the inverse? p "> a) [latex]p[/latex] : Dora is an explorer. b) [latex]q[/latex]: Boots is a monkey. d) Converse e) Converse Assume the conditional statement, [latex]p \rightarrow q[/latex]: “If Chadwick Boseman was an actor, then Chadwick Boseman did not star in the movie Black Panther  is false, and use it to answer the following questions. a) Write the converse of the statement in words and determine its truth value. The only way the conditional statement can be false is if the hypothesis, [latex]p[/latex]: Chadwick Boseman was an actor, is true and the conclusion, [latex]q[/latex]: Chadwick Boseman did not star in the movie Black Panther , is false. The converse is [latex]q \rightarrow p[/latex], and it is written in words as: “If Chadwick Boseman did not star in the movie Black Panther , then Chadwick Boseman was an actor.” This statement is true, because false [latex]\rightarrow[/latex]true is true. b) Write the inverse of the statement in words and determine its truth value. The inverse has the form “[latex]\sim{p} \rightarrow \sim{q}[/latex]. ” The written form is: “If Chadwick Boseman was not an actor, then Chadwick Boseman starred in the movie Black Panther .” Because [latex]p[/latex] is true, and [latex]q[/latex] is false, [latex]\sim{p}[/latex] is false, and [latex]\sim{q}[/latex] is true. This means the inverse is false [latex]\rightarrow[/latex] true, which is true. Alternatively, from Question 1, the converse is true, and because the inverse is logically equivalent to the converse it must also be true. c) Write the contrapositive of the statement in words and determine its truth value. The contrapositive is logically equivalent to the conditional. Because the conditional is false, the contrapositive is also false. This can be confirmed by looking at the truth values of the contrapositive statement. The contrapositive has the form “[latex]\sim{q}\rightarrow \sim{p}[/latex].” Because [latex]q[/latex] is false and [latex]p[/latex] is true, [latex]\sim{q}[/latex] is true and ~ p ~ p "> [latex]\sim{p}[/latex] is false. Therefore, [latex]\sim{q}\rightarrow \sim{p}[/latex] is true [latex]\rightarrow[/latex] false, which is false. The written form of the contrapositive is “If Chadwick Boseman starred in the movie Black Panther , then Chadwick Boseman was not an actor.” Assume the conditional statement [latex]p \rightarrow q[/latex]: “If my friend lives in Shreveport, then my friend does not live in Louisiana” is false, and use it to answer the following questions. a) If my friend does not live in Louisiana, then my friend lives in Shreveport. True. b) If my friend does not live in Shreveport, then my friend lives in Louisiana. True. c) If my friend lives in Louisiana, then my friend does not live in Shreveport. False. two expressions that have the same truth value for all possible combinations of truth claues for all variables in the expressions. A logical statement that is always true. the statement formed by interchanging the hypothesis and the conclusion of a conditional statement. The multiplicative inverse of a matrix the statement formed by interchanging and negating both the hypothesis and the conclusion of a conditional statement. Finite Mathematics Copyright © 2024 by LOUIS: The Louisiana Library Network is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License , except where otherwise noted. Share This Book Mathematicians Math Lessons Square Roots Math Calculators Contrapositive Definition Geometry – Understanding Logical Statements in Math The contrapositive in geometry is a logical relationship between statements that plays a crucial role in proofs and reasoning. In essence, for any given conditional statement “If ( p ), then ( q )”, the contrapositive is expressed as “If not ( q ), then not ( p )”. This forms the foundation for establishing the validity of statements within the realm of geometry, where logical consistency is key. This concept is not only pivotal for theoretical aspects but is also practical in geometric proofs, where establishing the truth of a converse by demonstrating the contrapositive can be an effective strategy. I hope you feel the same excitement I do diving into the logical nuances that shape our understanding of geometry. What is the Contrapositive Definition in Geometry In geometry, the concept of contrapositive takes a significant role in the realm of logic and proofs . When considering conditional statements, which often come in the form of “if-then” clauses, understanding the contrapositive is crucial. Let’s dissect a conditional statement . It typically has the form “If $p$, then $q$,” represented as $p \rightarrow q$. The contrapositive of this statement switches and negates both the hypothesis and the conclusion, resulting in “If not $q$, then not $p$,” which is denoted as $\neg q \rightarrow \neg p$. It’s a common misconception that contrapositive is complex, but with practice, one can easily grasp its usage. Contrapositives hold a unique characteristic: they always share the same truth value as the original conditional statement. This means if the original statement is true , so is the contrapositive , and vice versa, making them logically equivalent . This property is frequently used in mathematical theorems and their proofs , serving as a fundamental tool for mathematicians. Unlike the contrapositive , the converse (If $q$, then $p$) or the inverse (If not $p$, then not $q$) might not necessarily have the same truth value as the original statement, making them less reliable in proof logic. Here’s a table to summarize the relationships: Statement Type Format Conditional $p \rightarrow q$ Contrapositive $\neg q \rightarrow \neg p$ Converse $q \rightarrow p$ Inverse $\neg p \rightarrow \neg q$ Understanding and identifying these equivalences are part of my approach to ensure content accuracy and improve the practice preview in my teachings. Moreover, applying this knowledge enhances both logic skills and mathematical rigor. Geometry and Conditional Statements In my study of geometry , I often encounter various conditional statements that serve as the building blocks for formulating theorems and proofs .  These conditionals are expressed in the form “if p, then q,” denoted as $p \rightarrow q$. They are essential for understanding how different geometric concepts relate to each other. For instance, consider a statement involving congruent angles : If two angles are congruent , then they have the same measure . This can be written as $ \angle A \cong \angle B \rightarrow m\angle A = m\angle B$, where $m\angle $ represents the measure of an angle . To craft proofs , I also make use of other forms of statements derived from the original conditional, such as its contrapositive , which swaps and negates both the hypothesis and the conclusion and is logically equivalent to the original statement. The contrapositive is expressed as $\sim q \rightarrow \sim p$ or, in the context of our previous example, $m\angle A \neq m\angle B \rightarrow \angle A \ncong \angle B $. Statement Example Conditional $ \angle A \cong \angle B \rightarrow m\angle A = m\angle B $ Contrapositive $ m\angle A \neq m\angle B \rightarrow \angle A \ncong \angle B$ While exploring biconditional statements, I learned they are true if and only if both the original condition and its converse are true. They are denoted as ( p \leftrightarrow q ) and in our angle scenario, it reads as: two angles are congruent if and only if they have the same measure , written as ( \angle A \cong \angle B \leftrightarrow m\angle A = m\angle B ). Truth tables become a reliable tool in evaluating the validity of these statements. They exhaustively list all possible truth values that the hypothesis and conclusion can have, and what the resulting truth value of the statement would be. Understanding these relationships is crucial to my journey in geometry , as they aid me in deducing properties about figures like quadrilaterals and forming robust proofs about their characteristics. In exploring the concept of the contrapositive in geometry, I’ve delved into the logical framework that supports mathematical reasoning. A contrapositive rephrases a conditional statement by reversing and negating both the hypothesis and the conclusion. This form preserves the truth value. If the original conditional is $p \rightarrow q$ , the contrapositive is $\neg q \rightarrow \neg p$ , signifying that if $q$ is false, then $p$ must also be false. This fundamental aspect of logic ensures that if the original statement is true, so is the contrapositive . Unveiling this equivalence is a powerful tool in proofs and problem-solving. For instance, when I come across complex geometric proofs, understanding that the contrapositive of a given conditional is true if the original condition is true can provide alternate pathways to a solution. In geometry, grasping the contrapositive , along with its counterparts—the converse and the inverse , sharpens my ability to dissect and understand statements, and provides a strong foundation for constructing and validating geometric arguments. Armed with this knowledge, I’m well-equipped to tackle the logical structure of geometrical statements and their implications rigorously and clearly. Pre Calculus Probability Sets & Set Theory Trigonometry 346 IMAGES Logic Example: Negating a Conditional Statement Logic Example: Negating a Conditional Statement PPT PPT Conditional Statements PPT VIDEO Conditional Statements Hypothesis and conclusion Identifying Hypothesis and Conclusion of “If-Then” Statement Lesson 2 Section 2 Conditional Statements HYPOTHESIS STATEMENT IS ACCEPTED OR REJECTED l THESIS TIPS & GUIDE How to State the Hypothesis (Conditional Statements) Lesson 2-1 (part 1) Conditional Statements COMMENTS 2.2 Conditional Statements Flashcards A statement formed by interchanging the hypothesis and the conclusion in a conditional statement. ... A new statement formed by negating both the hypothesis and the conclusion. contrapositive of a conditional. Formed by exchanging the hypothesis and the conclusion and negating both of them. If two angles are adjacent, then they have the same ... Conditional Statements (15+ Examples in Geometry) So the converse is found by rearranging the hypothesis and conclusion, as Math Planet accurately states. Converse: "If yesterday was Tuesday, then today is Wednesday." What is the Inverse of a Statement? Now the inverse of an If-Then statement is found by negating (making negative) both the hypothesis and conclusion of the conditional ... Converse, Inverse & Contrapositive of Conditional Statement The Contrapositive of a Conditional Statement. Suppose you have the conditional statement [latex]{\color{blue}p} \to {\color{red}q}[/latex], we compose the contrapositive statement by interchanging the hypothesis and conclusion of the inverse of the same conditional statement.. In other words, to find the contrapositive, we first find the inverse of the given conditional statement then swap ... If-then statement (Geometry, Proof) If both statements are true or if both statements are false then the converse is true. A conditional and its converse do not mean the same thing. If we negate both the hypothesis and the conclusion we get a inverse statement: if a population do not consist of 50% men then the population do not consist of 50% women. $$\sim p\rightarrow \: \sim q$$ Conditional Statements A conditional statement is a statement in the form "If P, then Q," where P and Q are called the hypothesis and conclusion, respectively. ... The contrapositive of a conditional statement is formed by negating both the hypothesis and conclusion and flipping the order in which they appear. For example, the contrapositive of the statement ... Conditional Statements What is converse inverse and contrapositive? The inverse of a conditional statement retains order of hypothesis and conclusion while negating both the hypothesis and the conclusion. The converse ... Conditional Statement: Definition, Truth Table, Examples There are two parts of conditional statements, hypothesis and conclusion. The hypothesis or condition will begin with the "if" part, and the conclusion or action will begin with the "then" part. A conditional statement is also called "implication." Conditional Statements Examples: Example 1: If it is Sunday, then you can go to play. 1.1: Statements and Conditional Statements Suppose that Ed has exactly $52 in his wallet. The following four statements will use the four possible truth combinations for the hypothesis and conclusion of a conditional statement. If Ed has exactly $52 in his wallet, then he has $20 in his wallet. This is a true statement. Notice that both the hypothesis and the conclusion are true. Converse Meaning in Geometry The converse in geometry refers to a form of statement that arises when the hypothesis and conclusion of a conditional statement are switched. ... Contrapositive statements, on the other hand, are formed by both negating and reversing the hypothesis and conclusion of the original conditional statement. Working with Conditional Statements and Negations Determining the Hypothesis and Conclusion of a Conditional Statement. Step 1: Consider the conditional statement, p → q, read as "if p then q ." Note that sometimes → is written as ⊃ . Step ... Understanding the Inverse Statement For example, consider the conditional statement: "If it is raining, then the ground is wet.". To form the inverse statement, we simply negate both the hypothesis and the conclusion of the original statement. In this case, the inverse statement would be: "If it is not raining, then the ground is not wet.". The key idea in the inverse ... Mathwords: Inverse of a Conditional Negating both the hypothesis and conclusion of a conditional statement. For example, the inverse of "If it is raining then the grass is wet" is "If it is not raining then the grass is not wet". Note: As in the example, a proposition may be true but its inverse may be false. See also. Contrapositive, converse, biconditional. PDF 2.1 Conditional Statements The _____ of a conditional statement is formed by negating (inserting "not") the ... The _____ of a conditional statement is formed by negating the hypothesis and the conclusion of the converse. ... When two statements are both true or both false, we say that they are logically 2.5 Equivalent Statements 1. Write the hypothesis of the conditional statement and label it with a p p. 2. Write the conclusion of the conditional statement and label it with a q q. 3. Identify the following statement as the converse, inverse, or contrapositive: "If Dora is not an explorer, then Boots is not a monkey.". Conditional Statement : Definition & Negation and Verifying Inverse: To write the inverse of a conditional statement, negate both the hypothesis and the conclusion. Contrapositive: To write the contrapositive, first write the converse and then negate both the hypothesis and the conclusion. When two statements are both true or both false, they are called equivalent statements. PDF Conditional Statements and Equivalence the statement formed by negating both the hypothesis and the ... the hypothesis and the conclusion of a conditional statement. • If an angle does not measure 50 degrees, then it is not an acute angle. Let's use a counterexample to figure out whether this is a true statement. PDF Review questions: (try these before class) The contrapositive of a conditional statement is formed by negating both the hypothesis and the conclusion, and then interchanging the resulting negations. In other words, the contrapositive negates and switches the parts of the sentence. Conditional P Q Negation ~ P Converse Q P Inverse ~P ~Q Contrapositive ~Q ~P Fact: If a conditional ... 7.5 Equivalent Statements The conclusion of a conditional statement is the phrase following the then. The word then is not included when stating the conclusion. The answer is: [latex]q[/latex]: Lassie likes to bark. ... the statement formed by interchanging and negating both the hypothesis and the conclusion of a conditional statement. Chapter 2 Definitions Flashcards The statement formed by exchanging the hypothesis and conclusion of a conditional statement. ... The statement formed by both exchanging and negating the hypothesis and conclusion of a conditional statement. logically equivalent statements. Statements that have the same truth value. deductive reasoning. Contrapositive Definition Geometry The contrapositive in geometry is a logical relationship between statements that plays a crucial role in proofs and reasoning.. In essence, for any given conditional statement "If ( p ), then ( q )", the contrapositive is expressed as "If not ( q ), then not ( p )".. This forms the foundation for establishing the validity of statements within the realm of geometry, where logical ... CONVERSE,INVERSE,CONTRAPOSITIVE Flashcards converse of a conditional. inverse of a conditional. contrapositive of a conditional. A statement formed by interchanging the hypothesis and the conclusion in a conditional statement. A new statement formed by negating both the hypothesis and the conclusion. Formed by exchanging the hypothesis and the conclusion and negating both of them. assignment essay homework paper phd report resume review speech thesis