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Introduction to Proofs: An Active Exploration of Mathematical Language

Jennifer Firkins Nordstrom

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Section 2.3 conditional statements, activity 2.3.1 . which type., activity 2.3.2 . relationship between universal and conditional., example 2.3.1 . universal conditional statement..

Translate the statement using quantifiers and variables, “If an integer is even then it is divisible by 2.” Answer . Let \(P(x)\) be “ \(x\) is even” and \(Q(x)\) be “ \(x\) is divisible by 2.” \(\forall x\in \mathbb{Z}, P(x)\rightarrow Q(x)\text{.}\)

Activity 2.3.3 . Translating to Universal Conditional.

Activity 2.3.4 . a geography conditional., activity 2.3.5 . a weather conditional., activity 2.3.6 . an argument conditional., activity 2.3.7 . a mathematical conditional., logical equivalences for conditionals..

• \(\displaystyle p\rightarrow q\equiv \neg p\ \vee q\)
• \(\displaystyle \neg(p\rightarrow q)\equiv p\ \wedge \neg q\)

Definition 2.3.2 .

• contrapositive : \(\neg q\rightarrow \neg p\text{;}\)
• converse : \(q\rightarrow p\text{;}\)
• inverse : \(\neg p\rightarrow \neg q\text{.}\)

Definition 2.3.3 .

Definition 2.3.4 ., activity 2.3.8 . writing contrapositives., activity 2.3.9 . more writing contrapositives., activity 2.3.10 . converse statements., converting an argument to a conditional statement..

Example 2.3.5 . Converting and Argument to a Conditional.

Example 2.3.6 . Converting and Argument to a Conditional.

Activity 2.3.11 . Checking Validity with Conditional.

Activity 2.3.12 . More Checking Validity with Conditional.

Exercises Exercises

• Given any positive real number \(r\text{,}\) the reciprocal of ___.
• For any real number \(r\text{,}\) if \(r\) is ___ then ___.
• If a real number \(r\) ___, then ___.
• Given any negative real number \(s\text{,}\) the cube root of ___.
• For any real number \(s\text{,}\) if \(s\) is ___, then ___.
• If a real number \(s\) ___, then ___.
• There are real numbers \(u\) and \(v\) with the property that \(u+v < v-u\text{.}\)
• There is a real number \(x\) such that \(x^2 < x\text{.}\)
• For all positive integers \(n\text{,}\) \(n^2\geq n\text{.}\)
• For all real numbers \(a\) and \(b\text{,}\) \(| a+b | \leq | a | + | b |\text{.}\)
• All nonzero real numbers ___.
• For all nonzero real numbers \(r\text{,}\) there is ___ for \(r\text{.}\)
• For all nonzero real numbers \(r\text{,}\) there is a real number \(s\) such that ___.
• \(\forall\) real numbers \(x\text{,}\) if \(x>3\) then \(x^2>9\text{.}\)
• \(\forall n\in \mathbb{Z}\text{,}\) if \(n\) is prime then \(n\) is odd or \(n=2\text{.}\)
• \(\forall\) integers \(n\text{,}\) if \(n\) is divisible by 6, then \(n\) is divisible by 2 and \(n\) is divisible by 3.

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• Conditional Statement

What Is A Conditional Statement?

In mathematics, we define statement as a declarative statement which may either be true or may be false. Often sentences that are mathematical in nature may not be a statement because we might not know what the variable represents. For example, 2x + 2 = 5. Now here we do not know what x represents thus if we substitute the value of x (let us consider that x = 3) i.e., 2 × 3 = 6. Therefore, it is a false statement. So, what is a conditional statement? In simple words, when through a statement we put a condition on something in return of something, we call it a conditional statement. For example, Mohan tells his friend that “if you do my homework, then I will pay you 50 dollars”. So what is happening here? Mohan is paying his friend 50 dollars but places a condition that if only he’s work will be completed by his friend. A conditional statement is made up of two parts. First, there is a hypothesis that is placed after “if” and before the comma and second is a conclusion that is placed after “then”. Here, the hypothesis will be “you do my homework” and the conclusion will be “I will pay you 50 dollars”. Now, this statement can either be true or may be false. We don’t know. 

A hypothesis is a part that is used after the 'if' and before the comma. This composes the first part of a conditional statement. For example, the statement, 'I help you get an A+ in math,' is a hypothesis because this phrase is coming in between the 'if' and the comma. So, now I hope you can spot the hypothesis in other examples of a conditional statement. Of course, you can. Here is a statement: 'If Miley gets a car, then Allie's dog will be trained,' the hypothesis here is, 'Miley gets a car.' For the statement, 'If Tom eats chocolate ice cream, then Luke eats double chocolate ice cream,' the hypothesis here is, 'Tom eats chocolate ice cream. Now it is time for you to try and locate the hypothesis for the statement, 'If the square is a rectangle, then the rectangle is a quadrilateral'?

A conclusion is a part that is used after “then”. This composes the second part of a conditional statement. For example, for the statement, “I help you get an A+ in math”, the conclusion will be “you will give me 50 dollars”. The next statement was “If Miley gets a car, then Allie's dog will be trained”, the conclusion here is Allie's dog will be trained. It is the same with the next statement and for every other conditional statement.   

How Do We Know If A Statement Is True or False? 

In mathematics, the best way we can know if a statement is true or false is by writing a mathematical proof. Before writing a proof, the mathematician must find if the statement is true or false that can be done with the help of exploration and then by finding the counterexample. Once the proof is discovered, the mathematician must communicate this discovery to those who speak the language of maths. 

Converse, Inverse, contrapositive, And Bi-conditional Statement

We usually use the term “converse” as a verb for talking and chatting and as a noun we use it to represent a brand of footwear. But in mathematics, we use it differently. Converse and inverse are the two terms that are a connected concept in the making of a conditional statement.

If we want to create the converse of a conditional statement, we just have to switch the hypothesis and the conclusion. To create the inverse of a conditional statement, we have to turn both the hypothesis and the conclusion to the negative. A contrapositive statement can be made if we first interchange the hypothesis and conclusion then make them both negative. In a bi-conditional statement, we use “if and only if” which means that the hypothesis is true only if the condition is true. For example, 

If you eat junk food, then you will gain weight is a conditional statement.

If you gained weight, then you ate junk food is a converse of a conditional statement.

If you do not eat junk food, then you will not gain weight is an inverse of a conditional statement.

If yesterday was not Monday, then today is not Tuesday is a contrapositive statement. 

Today is Tuesday if and only if yesterday was Monday is a bi-conventional statement.   

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A Conditional Statement Truth Table

In the table above, p→q will be false only if the hypothesis(p) will be true and the conclusion(q) will be false, or else p→q will be true. 

Conditional Statement Examples

Below, you can see some of the conditional statement examples.

Example 1) Given, P = I do my work; Q = I get the allowance

What does p→q represent?

Solution 1) In the sentence above, the hypothesis is “I do my work” and the conclusion is “ I get the allowance”. Therefore, the condition p→q represents the conditional statement, “If I do my work, then I get the allowance”. 

Example 2) Given, a = The sun is a ball of gas; b = 5 is a prime number. Write a→b in a sentence. 

Solution 2) The conditional statement a→b here is “if the sun is a ball of gas, then 5 is a prime number”.

FAQs on Conditional Statement

1. How many types of conditional statements are there?

There are basically 5 types of conditional statements.

If statement, if-else statement, nested if-else statement, if-else-if ladder, and switch statement are the basic types of conditional statements. If a function displays a statement or performs a function on the condition if the statement is true. If-else statement executes a block of code if the condition is true but if the condition is false, a new block of code is placed. The switch statement is a selection control mechanism that allows the value of a variable to change the control flow of a program. 

2. How are a conditional statement and a loop different from each other?

A conditional statement is sometimes used by a loop but a loop is of no use to a conditional statement. A conditional statement is basically a “yes” or a “no” i.e., if yes, then take the first path else take the second one. A loop is more like a cyclic chain starting from the start point i.e., if yes, then take path a, if no, take path b and it comes back to the start point. 

Conditional statement executes a statement based on a condition without causing any repetition. 

A loop executes a statement repeatedly. There are two loop variables i.e., for loop and while loop.

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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"

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Conditional Statements

Big Picture

“If-then” relationships have an important role in geometry. Many geometric statements are actually if-then statements, also called conditional statements.

Conditional Statement: A statement with a hypothesis followed by a conclusion. Can be written in “if-then” form.

Hypothesis: The first, or “if,” part of a conditional statement. An educated guess.

Conclusion: The second, or “then,” part of a conditional statement. The conclusion of a hypothesis.

Converse: A statement where the hypothesis and conclusion of a conditional statement are switched.

Negation (of a statement) : The opposite of the original statement.

Inverse: A statement where the hypothesis and conclusion of a conditional statement are negated.

Contrapositive: A statement where the hypothesis and conclusion of a conditional statement are exchanged and negated.

Biconditional Statements: A statement where the original and the converse are both true.

Compound Statement: Combination of two or more statements.

Conjunction:   A compound statement using the word “and.”

Disjunction: A compound statement using the word “or.”

Truth Value: The truth value of a statement is either true or false.

Geometry uses conditional statements that can be symbolically written as \(p \rightarrow q\) (read as “if , then”). “If” is the hypothesis , and “then” is the conclusion .

• The conclusion is sometimes written before the hypothesis.
• Does not always have to include the words “if” and “then.”
• IF the weather is nice, THEN I’ll wash the car.
• IF the weather is nice, I’ll wash the car. (“Then” is implied.)
• I’ll wash the car IF the weather is nice. (The hypothesis comes after the conclusion.)
• All equiangular triangles are equilateral. (“If” and “then” are implied.)
• Can be written as: IF the triangle is equiangular, THEN it is equilateral.

The conditional statement is false when the hypothesis is true and the conclusion is false.

• Only one counter example is needed to prove the conditional statement false.

If the hypothesis is false, the conditional statement is true regardless of whether the conclusion is true or not.

Related Conditionals

The converse of \(p \rightarrow q\) is \(q \rightarrow p\).

The negation of p is “not p,” written as \(\sim p\).

The inverse of \(p \rightarrow q\) is \(\sim p \rightarrow \sim q\).

The contrapositive of \(p \rightarrow q\) is \(\sim{q} \rightarrow \sim{p}\).

Law of Contrapositive

Law of Contrapositive: If p \(p \rightarrow q\) q is true and \(\sim q\) is given, then \(\sim p\) is true.

If the conditional statement is true, the converse and inverse may or may not be true. However, the contrapositive of a true statement is always true. The contrapositive is logically equivalent to the original conditional statement.

• The converse and inverse are also logically equivalent.

Page 1 of 2

Conditional Statements cont.

Biconditional statements.

Biconditional statement: If \(p \rightarrow q\) is true and \(q \rightarrow p\) is true, it can be written as \(p \leftrightarrow q\).

• The \(\leftrightarrow\) is read as “if and only if” (abbreviated as “iff”).
• The “if and only if” portion of biconditional statement can be implied.

Compound Statement

A compound statement is a combination of two or more statements. Let p and q each represent a statement.

• Conjunction: A compound statement that uses the word “and.” \(p \land q\)
• Disjunction: A compound statement that uses the word “or.” \(p \lor q\)

Truth Table

A truth table can be used to analyze logic problems. It displays the truth values, either true (T) or false (F), for a conditional statement or a compound statement depending on the truth values for the hypothesis and conclusion.

• Start truth tables with all possible combinations of truth values.      a. There are four possible combinations for two variables and eight possible combinations for three variables.
• Do any negations.
• Do any combinations in parenthesis.
• Complete the problem.

Conditional

p \(\rightarrow \) q

\(q \rightarrow p\)

Conjunction

\(p \wedge q\)

\(\sim p \rightarrow \sim q\)

Contrapositive

\(\sim q \rightarrow \sim p\)

Disjunction

\(p \vee q\)

Page 2 of 2

Conditional Statement

An "if ... then ..." statement. It has a hypothesis and a conclusion like this: if hypothesis then conclusion

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 might also like these tutorials:

• Introduction to Truth Tables, Statements, and Logical Connectives
• Truth Tables of Five (5) Common Logical Connectives or Operators

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17.6: Truth Tables: Conditional, Biconditional

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• David Lippman
• Pierce College via The OpenTextBookStore

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We discussed conditional statements earlier, in which we take an action based on the value of the condition. We are now going to look at another version of a conditional, sometimes called an implication, which states that the second part must logically follow from the first.

Conditional

A conditional is a logical compound statement in which a statement \(p\), called the antecedent, implies a statement \(q\), called the consequent.

A conditional is written as \(p \rightarrow q\) and is translated as "if \(p\), then \(q\)".

The English statement “If it is raining, then there are clouds is the sky” is a conditional statement. It makes sense because if the antecedent “it is raining” is true, then the consequent “there are clouds in the sky” must also be true.

Notice that the statement tells us nothing of what to expect if it is not raining; there might be clouds in the sky, or there might not. If the antecedent is false, then the consquent becomes irrelevant.

Suppose you order a team jersey online on Tuesday and want to receive it by Friday so you can wear it to Saturday’s game. The website says that if you pay for expedited shipping, you will receive the jersey by Friday. In what situation is the website telling a lie?

There are four possible outcomes:

1) You pay for expedited shipping and receive the jersey by Friday

2) You pay for expedited shipping and don’t receive the jersey by Friday

3) You don’t pay for expedited shipping and receive the jersey by Friday

4) You don’t pay for expedited shipping and don’t receive the jersey by Friday

Only one of these outcomes proves that the website was lying: the second outcome in which you pay for expedited shipping but don’t receive the jersey by Friday. The first outcome is exactly what was promised, so there’s no problem with that. The third outcome is not a lie because the website never said what would happen if you didn’t pay for expedited shipping; maybe the jersey would arrive by Friday whether you paid for expedited shipping or not. The fourth outcome is not a lie because, again, the website didn’t make any promises about when the jersey would arrive if you didn’t pay for expedited shipping.

It may seem strange that the third outcome in the previous example, in which the first part is false but the second part is true, is not a lie. Remember, though, that if the antecedent is false, we cannot make any judgment about the consequent. The website never said that paying for expedited shipping was the only way to receive the jersey by Friday.

A friend tells you “If you upload that picture to Facebook, you’ll lose your job.” Under what conditions can you say that your friend was wrong?

1) You upload the picture and lose your job

2) You upload the picture and don’t lose your job

3) You don’t upload the picture and lose your job

4) You don’t upload the picture and don’t lose your job

There is only one possible case in which you can say your friend was wrong: the second outcome in which you upload the picture but still keep your job. In the last two cases, your friend didn’t say anything about what would happen if you didn’t upload the picture, so you can’t say that their statement was wrong. Even if you didn’t upload the picture and lost your job anyway, your friend never said that you were guaranteed to keep your job if you didn’t upload the picture; you might lose your job for missing a shift or punching your boss instead.

In traditional logic, a conditional is considered true as long as there are no cases in which the antecedent is true and the consequent is false.

Truth table for the conditional

\(\begin{array}{|c|c|c|} \hline p & q & p \rightarrow q \\ \hline \mathrm{T} & \mathrm{T} & \mathrm{T} \\ \hline \mathrm{T} & \mathrm{F} & \mathrm{F} \\ \hline \mathrm{F} & \mathrm{T} & \mathrm{T} \\ \hline \mathrm{F} & \mathrm{F} & \mathrm{T} \\ \hline \end{array}\)

Again, if the antecedent \(p\) is false, we cannot prove that the statement is a lie, so the result of the third and fourth rows is true.

Construct a truth table for the statement \((m \wedge \sim p) \rightarrow r\)

We start by constructing a truth table with 8 rows to cover all possible scenarios. Next, we can focus on the antecedent, \(m \wedge \sim p\).

\(\begin{array}{|c|c|c|} \hline m & p & r \\ \hline \mathrm{T} & \mathrm{T} & \mathrm{T} \\ \hline \mathrm{T} & \mathrm{T} & \mathrm{F} \\ \hline \mathrm{T} & \mathrm{F} & \mathrm{T} \\ \hline \mathrm{T} & \mathrm{F} & \mathrm{F} \\ \hline \mathrm{F} & \mathrm{T} & \mathrm{T} \\ \hline \mathrm{F} & \mathrm{T} & \mathrm{F} \\ \hline \mathrm{F} & \mathrm{F} & \mathrm{T} \\ \hline \mathrm{F} & \mathrm{F} & \mathrm{F} \\ \hline \end{array}\)

\(\begin{array}{|c|c|c|c|} \hline m & p & r & \sim p \\ \hline \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{F} \\ \hline \mathrm{T} & \mathrm{T} & \mathrm{F} & \mathrm{F} \\ \hline \mathrm{T} & \mathrm{F} & \mathrm{T} & \mathrm{T} \\ \hline \mathrm{T} & \mathrm{F} & \mathrm{F} & \mathrm{T} \\ \hline \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{F} \\ \hline \mathrm{F} & \mathrm{T} & \mathrm{F} & \mathrm{F} \\ \hline \mathrm{F} & \mathrm{F} & \mathrm{T} & \mathrm{T} \\ \hline \mathrm{F} & \mathrm{F} & \mathrm{F} & \mathrm{T} \\ \hline \end{array}\)

\(\begin{array}{|c|c|c|c|c|} \hline m & p & r & \sim p & m \wedge \sim p \\ \hline \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{F} & \mathrm{F} \\ \hline \mathrm{T} & \mathrm{T} & \mathrm{F} & \mathrm{F} & \mathrm{F} \\ \hline \mathrm{T} & \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{T} \\ \hline \mathrm{T} & \mathrm{F} & \mathrm{F} & \mathrm{T} & \mathrm{T} \\ \hline \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{F} & \mathrm{F} \\ \hline \mathrm{F} & \mathrm{T} & \mathrm{F} & \mathrm{F} & \mathrm{F} \\ \hline \mathrm{F} & \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{F} \\ \hline \mathrm{F} & \mathrm{F} & \mathrm{F} & \mathrm{T} & \mathrm{F} \\ \hline \end{array}\)

Now we can create a column for the conditional. Because it can be confusing to keep track of all the Ts and \(\mathrm{Fs}\), why don't we copy the column for \(r\) to the right of the column for \(m \wedge \sim p\) ? This makes it a lot easier to read the conditional from left to right.

\(\begin{array}{|c|c|c|c|c|c|c|} \hline m & p & r & \sim p & m \wedge \sim p & r & (m \wedge \sim p) \rightarrow r \\ \hline \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{F} & \mathrm{F} & \mathrm{T} & \mathrm{T} \\ \hline \mathrm{T} & \mathrm{T} & \mathrm{F} & \mathrm{F} & \mathrm{F} & \mathrm{F} & \mathrm{T} \\ \hline \mathrm{T} & \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} \\ \hline \mathrm{T} & \mathrm{F} & \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{F} & \mathrm{F} \\ \hline \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{F} & \mathrm{F} & \mathrm{T} & \mathrm{T} \\ \hline \mathrm{F} & \mathrm{T} & \mathrm{F} & \mathrm{F} & \mathrm{F} & \mathrm{F} & \mathrm{T} \\ \hline \mathrm{F} & \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{F} & \mathrm{T} & \mathrm{T} \\ \hline \mathrm{F} & \mathrm{F} & \mathrm{F} & \mathrm{T} & \mathrm{F} & \mathrm{F} & \mathrm{T} \\ \hline \end{array}\)

When \(m\) is true, \(p\) is false, and \(r\) is false- -the fourth row of the table-then the antecedent \(m \wedge \sim p\) will be true but the consequent false, resulting in an invalid conditional; every other case gives a valid conditional.

If you want a real-life situation that could be modeled by \((m \wedge \sim p) \rightarrow r\), consider this: let \(m=\) we order meatballs, \(p=\) we order pasta, and \(r=\) Rob is happy. The statement \((m \wedge \sim p) \rightarrow r\) is "if we order meatballs and don't order pasta, then Rob is happy". If \(m\) is true (we order meatballs), \(p\) is false (we don't order pasta), and \(r\) is false (Rob is not happy), then the statement is false, because we satisfied the antecedent but Rob did not satisfy the consequent.

For any conditional, there are three related statements, the converse, the inverse, and the contrapositive.

Related Statments

The original conditional is \(\quad\) "if \(p,\) then \(q^{\prime \prime} \quad p \rightarrow q\)

The converse is \(\quad\) "if \(q,\) then \(p^{\prime \prime} \quad q \rightarrow p\)

The inverse is \(\quad\) "if not \(p,\) then not \(q^{\prime \prime} \quad \sim p \rightarrow \sim q\)

The contrapositive is "if not \(q,\) then not \(p^{\prime \prime} \quad \sim q \rightarrow \sim p\)

Consider again the conditional “If it is raining, then there are clouds in the sky.” It seems reasonable to assume that this is true.

The converse would be “If there are clouds in the sky, then it is raining.” This is not always true.

The inverse would be “If it is not raining, then there are not clouds in the sky.” Likewise, this is not always true.

The contrapositive would be “If there are not clouds in the sky, then it is not raining.” This statement is true, and is equivalent to the original conditional.

Looking at truth tables, we can see that the original conditional and the contrapositive are logically equivalent, and that the converse and inverse are logically equivalent.

Equivalence

A conditional statement and its contrapositive are logically equivalent.

The converse and inverse of a conditional statement are logically equivalent.

In other words, the original statement and the contrapositive must agree with each other; they must both be true, or they must both be false. Similarly, the converse and the inverse must agree with each other; they must both be true, or they must both be false.

Be aware that symbolic logic cannot represent the English language perfectly. For example, we may need to change the verb tense to show that one thing occurred before another.

Suppose this statement is true: “If I eat this giant cookie, then I will feel sick.” Which of the following statements must also be true?

• If I feel sick, then I ate that giant cookie.
• If I don’t eat this giant cookie, then I won’t feel sick.
• If I don’t feel sick, then I didn’t eat that giant cookie.
• This is the converse, which is not necessarily true. I could feel sick for some other reason, such as drinking sour milk.
• This is the inverse, which is not necessarily true. Again, I could feel sick for some other reason; avoiding the cookie doesn’t guarantee that I won’t feel sick.
• This is the contrapositive, which is true, but we have to think somewhat backwards to explain it. If I ate the cookie, I would feel sick, but since I don’t feel sick, I must not have eaten the cookie.

Notice again that the original statement and the contrapositive have the same truth value (both are true), and the converse and the inverse have the same truth value (both are false).

Try it Now 5

“If you microwave salmon in the staff kitchen, then I will be mad at you.” If this statement is true, which of the following statements must also be true?

• If you don’t microwave salmon in the staff kitchen, then I won’t be mad at you.
• If I am not mad at you, then you didn’t microwave salmon in the staff kitchen.
• If I am mad at you, then you microwaved salmon in the staff kitchen.

Choice b is correct because it is the contrapositive of the original statement.

Consider the statement “If you park here, then you will get a ticket.” What set of conditions would prove this statement false?

• You don’t park here and you get a ticket.
• You don’t park here and you don’t get a ticket.
• You park here and you don’t get a ticket.

The first two statements are irrelevant because we don’t know what will happen if you park somewhere else. The third statement, however contradicts the conditional statement “If you park here, then you will get a ticket” because you parked here but didn’t get a ticket. This example demonstrates a general rule; the negation of a conditional can be written as a conjunction: “It is not the case that if you park here, then you will get a ticket” is equivalent to “You park here and you do not get a ticket.”

The Negation of a Conditional

The negation of a conditional statement is logically equivalent to a conjunction of the antecedent and the negation of the consequent.

\(\sim(p \rightarrow q)\) is equivalent to \(p \wedge \sim q\)

Which of the following statements is equivalent to the negation of “If you don’t grease the pan, then the food will stick to it” ?

• I didn’t grease the pan and the food didn’t stick to it.
• I didn’t grease the pan and the food stuck to it.
• I greased the pan and the food didn’t stick to it.
• This is correct; it is the conjunction of the antecedent and the negation of the consequent. To disprove that not greasing the pan will cause the food to stick, I have to not grease the pan and have the food not stick.
• This is essentially the original statement with no negation; the “if…then” has been replaced by “and”.
• This essentially agrees with the original statement and cannot disprove it.

Try it Now 6

“If you go swimming less than an hour after eating lunch, then you will get cramps.” Which of the following statements is equivalent to the negation of this statement?

• I went swimming more than an hour after eating lunch and I got cramps.
• I went swimming less than an hour after eating lunch and I didn’t get cramps.
• I went swimming more than an hour after eating lunch and I didn’t get cramps.

Choice b is equivalent to the negation; it keeps the first part the same and negates the second part.

In everyday life, we often have a stronger meaning in mind when we use a conditional statement. Consider “If you submit your hours today, then you will be paid next Friday.” What the payroll rep really means is “If you submit your hours today, then you will be paid next Friday, and if you don’t submit your hours today, then you won’t be paid next Friday.” The conditional statement if t , then p also includes the inverse of the statement: if not t , then not p . A more compact way to express this statement is “You will be paid next Friday if and only if you submit your timesheet today.” A statement of this form is called a biconditional .

Biconditional

A biconditional is a logical conditional statement in which the antecedent and consequent are interchangeable.

A biconditional is written as \(p \leftrightarrow q\) and is translated as " \(p\) if and only if \(q^{\prime \prime}\).

Because a biconditional statement \(p \leftrightarrow q\) is equivalent to \((p \rightarrow q) \wedge(q \rightarrow p),\) we may think of it as a conditional statement combined with its converse: if \(p\), then \(q\) and if \(q\), then \(p\). The double-headed arrow shows that the conditional statement goes from left to right and from right to left. A biconditional is considered true as long as the antecedent and the consequent have the same truth value; that is, they are either both true or both false.

Truth table for the biconditional

\(\begin{array}{|c|c|c|} \hline p & q & p \leftrightarrow q \\ \hline \mathrm{T} & \mathrm{T} & \mathrm{T} \\ \hline \mathrm{T} & \mathrm{F} & \mathrm{F} \\ \hline \mathrm{F} & \mathrm{T} & \mathrm{F} \\ \hline \mathrm{F} & \mathrm{F} & \mathrm{T} \\ \hline \end{array}\)

Notice that the fourth row, where both components are false, is true; if you don’t submit your timesheet and you don’t get paid, the person from payroll told you the truth.

Suppose this statement is true: “The garbage truck comes down my street if and only if it is Thursday morning.” Which of the following statements could be true?

• It is noon on Thursday and the garbage truck did not come down my street this morning.
• It is Monday and the garbage truck is coming down my street.
• It is Wednesday at 11:59PM and the garbage truck did not come down my street today.
• This cannot be true. This is like the second row of the truth table; it is true that I just experienced Thursday morning, but it is false that the garbage truck came.
• This cannot be true. This is like the third row of the truth table; it is false that it is Thursday, but it is true that the garbage truck came.
• This could be true. This is like the fourth row of the truth table; it is false that it is Thursday, but it is also false that the garbage truck came, so everything worked out like it should.

Try it Now 7

Suppose this statement is true: “I wear my running shoes if and only if I am exercising.” Determine whether each of the following statements must be true or false.

• I am exercising and I am not wearing my running shoes.
• I am wearing my running shoes and I am not exercising.
• I am not exercising and I am not wearing my running shoes.

Choices a & b are false; c is true.

Create a truth table for the statement \((A \vee B) \leftrightarrow \sim C\)

Whenever we have three component statements, we start by listing all the possible truth value combinations for \(A, B,\) and \(C .\) After creating those three columns, we can create a fourth column for the antecedent, \(A \vee B\). Now we will temporarily ignore the column for \(C\) and focus on \(A\) and \(B\), writing the truth values for \(A \vee B\).

\(\begin{array}{|c|c|c|} \hline A & B & C \\ \hline \mathrm{T} & \mathrm{T} & \mathrm{T} \\ \hline \mathrm{T} & \mathrm{T} & \mathrm{F} \\ \hline \mathrm{T} & \mathrm{F} & \mathrm{T} \\ \hline \mathrm{T} & \mathrm{F} & \mathrm{F} \\ \hline \mathrm{F} & \mathrm{T} & \mathrm{T} \\ \hline \mathrm{F} & \mathrm{T} & \mathrm{F} \\ \hline \mathrm{F} & \mathrm{F} & \mathrm{T} \\ \hline \mathrm{F} & \mathrm{F} & \mathrm{F} \\ \hline \end{array}\)

\(\begin{array}{|c|c|c|c|} \hline A & B & C & A \vee B \\ \hline \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} \\ \hline \mathrm{T} & \mathrm{T} & \mathrm{F} & \mathrm{T} \\ \hline \mathrm{T} & \mathrm{F} & \mathrm{T} & \mathrm{T} \\ \hline \mathrm{T} & \mathrm{F} & \mathrm{F} & \mathrm{T} \\ \hline \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{T} \\ \hline \mathrm{F} & \mathrm{T} & \mathrm{F} & \mathrm{T} \\ \hline \mathrm{F} & \mathrm{F} & \mathrm{T} & \mathrm{F} \\ \hline \mathrm{F} & \mathrm{F} & \mathrm{F} & \mathrm{F} \\ \hline \end{array}\)

Next we can create a column for the negation of \(C\). (Ignore the \(A \vee B\) column and simply negate the values in the \(C\) column.)

\(\begin{array}{|c|c|c|c|c|} \hline A & B & C & A \vee B & \sim C \\ \hline \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{F} \\ \hline \mathrm{T} & \mathrm{T} & \mathrm{F} & \mathrm{T} & \mathrm{T} \\ \hline \mathrm{T} & \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{F} \\ \hline \mathrm{T} & \mathrm{F} & \mathrm{F} & \mathrm{T} & \mathrm{T} \\ \hline \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{F} \\ \hline \mathrm{F} & \mathrm{T} & \mathrm{F} & \mathrm{T} & \mathrm{T} \\ \hline \mathrm{F} & \mathrm{F} & \mathrm{T} & \mathrm{F} & \mathrm{F} \\ \hline \mathrm{F} & \mathrm{F} & \mathrm{F} & \mathrm{F} & \mathrm{T} \\ \hline \end{array}\)

Finally, we find the truth values of \((A \vee B) \leftrightarrow \sim C\). Remember, a biconditional is true when the truth value of the two parts match, but it is false when the truth values do not match.

\(\begin{array}{|c|c|c|c|c|c|} \hline A & B & C & A \vee B & \sim C & (A \vee B) \leftrightarrow \sim C \\ \hline \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{F} & \mathrm{F} \\ \hline \mathrm{T} & \mathrm{T} & \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{T} \\ \hline \mathrm{T} & \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{F} & \mathrm{F} \\ \hline \mathrm{T} & \mathrm{F} & \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{T} \\ \hline \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{F} & \mathrm{F} \\ \hline \mathrm{F} & \mathrm{T} & \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{T} \\ \hline \mathrm{F} & \mathrm{F} & \mathrm{T} & \mathrm{F} & \mathrm{F} & \mathrm{T} \\ \hline \mathrm{F} & \mathrm{F} & \mathrm{F} & \mathrm{F} & \mathrm{T} & \mathrm{F} \\ \hline \end{array}\)

To illustrate this situation, suppose your boss needs you to do either project \(A\) or project \(B\) (or both, if you have the time). If you do one of the projects, you will not get a crummy review ( \(C\) is for crummy). So \((A \vee B) \leftrightarrow \sim C\) means "You will not get a crummy review if and only if you do project \(A\) or project \(B\)." Looking at a few of the rows of the truth table, we can see how this works out. In the first row, \(A, B,\) and \(C\) are all true: you did both projects and got a crummy review, which is not what your boss told you would happen! That is why the final result of the first row is false. In the fourth row, \(A\) is true, \(B\) is false, and \(C\) is false: you did project \(A\) and did not get a crummy review. This is what your boss said would happen, so the final result of this row is true. And in the eighth row, \(A, B\), and \(C\) are all false: you didn't do either project and did not get a crummy review. This is not what your boss said would happen, so the final result of this row is false. (Even though you may be happy that your boss didn't follow through on the threat, the truth table shows that your boss lied about what would happen.)

Conditional Statements

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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.

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Conditional Statements

Two common types of statements found in the study of logic are conditional and biconditional statements. They are formed by combining two statements which we then we call compound statements. What if we were to say, "If it snows, then we don't go outside." This is two statements combined. They are often called if-then statements. As in, "IF it snows, THEN we don't go outside." They are a fundamental building block of computer programming.

Writing conditional statements

A statement written in if-then format is a conditional statement.

It looks like

This represents the conditional statement:

"If p then q ."

A conditional statement is also called an implication.

If a closed shape has three sides, then it is a triangle.

The part of the statement that follows the "if" is called the hypothesis. The part of the statement that follows the "then" is the conclusion.

So in the above statement,

If a closed shape has three sides, (this is the hypothesis)

Then it is a triangle. (this is the conclusion)

Identify the hypothesis and conclusion of the following conditional statement.

A polygon is a hexagon if it has six sides.

Hypothesis: The polygon has six sides.

Conclusion: It is a hexagon.

The hypothesis does not always come first in a conditional statement. You must read it carefully to determine which part of the statement is the hypothesis and which part is the conclusion.

Truth table for conditional statement

The truth table for any two given inputs, say A and B , is given by:

• If A and B are both true, then A → B is true.
• If A is true and B is false, then A → B is false.
• If A is false and B is true, then A → B is true.
• If A and B are both false, then A → B is true.

Take our conditional statement that if it snows, we do not go outside.

If it is snowing ( A is true) and we do go outside ( B is false), then the statement A → B is false.

If it is not snowing ( A is false), it doesn't matter if we go outside or not ( B is true or false), because A → B is impossible to determine if A is false, so the statement A → B can still be true.

Biconditional statements

A biconditional statement is a combination of a statement and its opposite written in the format of "if and only if."

For example, "Two line segments are congruent if and only if they are the same length."

This is a combination of two conditional statements.

"Two line segments are congruent if they are the same length."

"Two line segments are the same length if they are congruent."

A biconditional statement is true if and only if both the conditional statements are true.

Biconditional statements are represented by the symbol:

p ↔ q

p ↔ q = p → q ∧ q → p

Writing biconditional statements

Write the two conditional statements that make up this biconditional statement:

I am punctual if and only if I am on time to school every day.

The two conditional statements that have to be true to make this statement true are:

• I am punctual if I am on time to school every day.
• I am on time to school every day if I am punctual.

A rectangle is a square if and only if the adjacent sides are congruent.

• If the adjacent sides of a rectangle are congruent then it is a square.
• If a rectangle is a square then the adjacent sides are congruent.

Topics related to the Conditional Statements

Conjunction

Counter Example

Biconditional Statement

Flashcards covering the Conditional Statements

Symbolic Logic Flashcards

Introduction to Proofs Flashcards

Practice tests covering the Conditional Statements

Introduction to Proofs Practice Tests

Get help learning about conditional statements

Understanding conditional statements can be tricky, especially when it gets deeper into programming language. If your student needs a boost in their comprehension of conditional statements, have them meet with an expert tutor who can give them 1-on-1 support in a setting free from distractions. A tutor can work at your student's pace so that tutoring is efficient while using their learning style - so that tutoring is effective. To learn more about how tutoring can help your student master conditional statements, contact the Educational Directors at Varsity Tutors today.

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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

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