problem solving with 3 sets

1.5 Set Operations with Three Sets

Learning objectives.

After completing this section, you should be able to:

• Interpret Venn diagrams with three sets.
• Create Venn diagrams with three sets.
• Apply set operations to three sets.
• Prove equality of sets using Venn diagrams.

Have you ever searched for something on the Internet and then soon after started seeing multiple advertisements for that item while browsing other web pages? Large corporations have built their business on data collection and analysis. As we start working with larger data sets, the analysis becomes more complex. In this section, we will extend our knowledge of set relationships by including a third set.

A Venn diagram with two intersecting sets breaks up the universal set into four regions; simply adding one additional set will increase the number of regions to eight, doubling the complexity of the problem.

Venn Diagrams with Three Sets

Below is a Venn diagram with two intersecting sets, which breaks the universal set up into four distinct regions.

Next, we see a Venn diagram with three intersecting sets , which breaks up the universal set into eight distinct regions.

Shading Venn Diagrams

Venn Diagram is an Android application that allows you to visualize how the sets are related in a Venn diagram by entering expressions and displaying the resulting Venn diagram of the set shaded in gray.

The Venn Diagram application uses some notation that differs from the notation covered in this text.

• The complement of set A A in this text is written symbolically as A ′ A ′ , but the Venn Diagram app uses A C A C to represent the complement operation.
• The set difference operation, − − , is available in the Venn Diagram app, although this operation is not covered in the text.

It is recommended that you explore this application to expand your knowledge of Venn diagrams prior to continuing with the next example.

In the next example, we will explore the three main blood factors, A, B and Rh. The following background information about blood types will help explain the relationships between the sets of blood factors. If an individual has blood factor A or B, those will be included in their blood type. The Rh factor is indicated with a + + or a − − . For example, if a person has all three blood factors, then their blood type would be AB + AB + . In the Venn diagram, they would be in the intersection of all three sets, A ∩ B ∩ R h + . A ∩ B ∩ R h + . If a person did not have any of these three blood factors, then their blood type would be O − , O − , and they would be in the set ( A ∪ B ∪ R h + ) ′ ( A ∪ B ∪ R h + ) ′ which is the region outside all three circles.

Example 1.35

Interpreting a venn diagram with three sets.

Use the Venn diagram below, which shows the blood types of 100 people who donated blood at a local clinic, to answer the following questions.

• How many people with a type A blood factor donated blood?
• Julio has blood type B + . B + . If he needs to have surgery that requires a blood transfusion, he can accept blood from anyone who does not have a type A blood factor. How many people donated blood that Julio can accept?
• How many people who donated blood do not have the Rh + Rh + blood factor?
• How many people had type A and type B blood?
• The number of people who donated blood with a type A blood factor will include the sum of all the values included in the A circle. It will be the union of sets A − , A + , A B − and A B + . A − , A + , A B − and A B + . n ( A ) = n ( A − ) + n ( A + ) + n ( A B − ) + n ( A B + ) = 6 + 36 + 1 + 3 = 46. n ( A ) = n ( A − ) + n ( A + ) + n ( A B − ) + n ( A B + ) = 6 + 36 + 1 + 3 = 46.
• In part 1, it was determined that the number of donors with a type A blood factor is 46. To determine the number of people who did not have a type A blood factor, use the following property, A ′ A ′ union is equal to U U , which means n ( A ) + n ( A ′ ) = n ( U ) , n ( A ) + n ( A ′ ) = n ( U ) , and n ( A ′ ) = n ( U ) − n ( A ) = 100 − 46 = 54. n ( A ′ ) = n ( U ) − n ( A ) = 100 − 46 = 54. Thus, 54 people donated blood that Julio can accept.
• This would be everyone outside the Rh + Rh + circle, or everyone with a negative Rh factor, n ( R h − ) = n ( O − ) + n ( A − ) + n ( A B − ) + n ( B − ) = 7 + 6 + 1 + 2 = 16. n ( R h − ) = n ( O − ) + n ( A − ) + n ( A B − ) + n ( B − ) = 7 + 6 + 1 + 2 = 16.
• To have both blood type A and blood type B, a person would need to be in the intersection of sets A A and B B . The two circles overlap in the regions labeled A B − A B − and A B + . A B + . Add up the number of people in these two regions to get the total: 1 + 3 = 4. 1 + 3 = 4. This can be written symbolically as n ( A and B ) = n ( A ∩ B ) = n ( A B − ) + n ( A B + ) = 1 + 3 = 4. n ( A and B ) = n ( A ∩ B ) = n ( A B − ) + n ( A B + ) = 1 + 3 = 4.

Your Turn 1.35

Blood types.

Most people know their main blood type of A, B, AB, or O and whether they are R h + R h + or R h − R h − , but did you know that the International Society of Blood Transfusion recognizes twenty-eight additional blood types that have important implications for organ transplants and successful pregnancy? For more information, check out this article:

Blood mystery solved: Two new blood types identified

Creating Venn Diagrams with Three Sets

In general, when creating Venn diagrams from data involving three subsets of a universal set, the strategy is to work from the inside out. Start with the intersection of the three sets, then address the regions that involve the intersection of two sets. Next, complete the regions that involve a single set, and finally address the region in the universal set that does not intersect with any of the three sets. This method can be extended to any number of sets. The key is to start with the region involving the most overlap, working your way from the center out.

Example 1.36

Creating a venn diagram with three sets.

A teacher surveyed her class of 43 students to find out how they prepared for their last test. She found that 24 students made flash cards, 14 studied their notes, and 27 completed the review assignment. Of the entire class of 43 students, 12 completed the review and made flash cards, nine completed the review and studied their notes, and seven made flash cards and studied their notes, while only five students completed all three of these tasks. The remaining students did not do any of these tasks. Create a Venn diagram with subsets labeled: “Notes,” “Flash Cards,” and “Review” to represent how the students prepared for the test.

Step 1: First, draw a Venn diagram with three intersecting circles to represent the three intersecting sets: Notes, Flash Cards, and Review. Label the universal set with the cardinality of the class.

Step 2: Next, in the region where all three sets intersect, enter the number of students who completed all three tasks.

Step 3: Next, calculate the value and label the three sections where just two sets overlap.

• Review and flash card overlap . A total of 12 students completed the review and made flash cards, but five of these twelve students did all three tasks, so we need to subtract: 12 − 5 = 7 12 − 5 = 7 . This is the value for the region where the flash card set intersects with the review set.
• Review and notes overlap . A total of 9 students completed the review and studied their notes, but again, five of these nine students completed all three tasks. So, we subtract: 9 − 5 = 4 9 − 5 = 4 . This is the value for the region where the review set intersects with the notes set.
• Flash card and notes overlap . A total of 7 students made flash cards and studied their notes; subtracting the five students that did all three tasks from this number leaves 2 students who only studied their notes and made flash cards. Add these values to the Venn diagram.

Step 4: Now, repeat this process to find the number of students who only completed one of these three tasks.

• A total of 24 students completed flash cards, but we have already accounted for 2 + 5 + 7 = 14 2 + 5 + 7 = 14 of these. Thus, 24 - 14 = 10 24 - 14 = 10 students who just made flash cards.
• A total of 14 students studied their notes, but we have already accounted for 4 + 5 + 2 = 11 4 + 5 + 2 = 11 of these. Thus, 14 - 11 = 3 14 - 11 = 3 students only studied their notes.
• A total of 27 students completed the review assignment, but we have already accounted for 4 + 5 + 7 = 16 4 + 5 + 7 = 16 of these, which means 27 - 16 = 11 27 - 16 = 11 students only completed the review assignment.
• Add these values to the Venn diagram.

Step 5: Finally, compute how many students did not do any of these three tasks. To do this, we add together each value that we have already calculated for the separate and intersecting sections of our three sets: 3 + 2 + 4 + 5 + 10 + 7 + 11 = 42 3 + 2 + 4 + 5 + 10 + 7 + 11 = 42 . Because there 43 students in the class, and 43 − 42 = 1 43 − 42 = 1 , this means only one student did not complete any of these tasks to prepare for the test. Record this value somewhere in the rectangle, but outside of all the circles, to complete the Venn diagram.

Your Turn 1.36

Applying set operations to three sets.

Set operations are applied between two sets at a time. Parentheses indicate which operation should be performed first. As with numbers, the inner most parentheses are applied first. Next, find the complement of any sets, then perform any union or intersections that remain.

Example 1.37

Perform the set operations as indicated on the following sets: U = { 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 } U = { 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 } , A = { 0 , 1 , 2 , 3 , 4 , 5 , 6 } , A = { 0 , 1 , 2 , 3 , 4 , 5 , 6 } , B = { 0 , 2 , 4 , 6 , 8 , 10 , 12 } , B = { 0 , 2 , 4 , 6 , 8 , 10 , 12 } , and C = { 0 , 3 , 6 , 9 , 12 } . C = { 0 , 3 , 6 , 9 , 12 } .

• Find ( A ∩ B ) ∩ C . ( A ∩ B ) ∩ C .
• Find A ∩ ( B ∪ C ) . A ∩ ( B ∪ C ) .
• Find ( A ∩ B ) ∪ C ′ . ( A ∩ B ) ∪ C ′ .
• Parentheses first, A A intersection B B equals A ∩ B = { 0 , 2 , 4 , 6 } , A ∩ B = { 0 , 2 , 4 , 6 } , the elements common to both A A and B B . ( A ∩ B ) ∩ C = { 0 , 2 , 4 , 6 } ∩ { 0 , 3 , 6 , 9 , 12 } = { 0 , 6 } , ( A ∩ B ) ∩ C = { 0 , 2 , 4 , 6 } ∩ { 0 , 3 , 6 , 9 , 12 } = { 0 , 6 } , because the only elements that are in both sets are 0 and 6.
• Parentheses first, B B union C C equals B ∪ C = { 0 , 2 , 3 , 4 , 6 , 8 , 9 , 10 , 12 } , B ∪ C = { 0 , 2 , 3 , 4 , 6 , 8 , 9 , 10 , 12 } , the collection of all elements in set B B or set C C or both. A ∩ ( B ∪ C ) = { 0 , 1 , 2 , 3 , 4 , 5 , 6 } ∩ { 0 , 2 , 3 , 4 , 6 , 8 , 9 , 10 , 12 } = { 0 , 2 , 3 , 4 , 6 } , A ∩ ( B ∪ C ) = { 0 , 1 , 2 , 3 , 4 , 5 , 6 } ∩ { 0 , 2 , 3 , 4 , 6 , 8 , 9 , 10 , 12 } = { 0 , 2 , 3 , 4 , 6 } , because the intersection of these two sets is the set of elements that are common to both sets.
• Parentheses first, A A intersection B B equals A ∩ B = { 0 , 2 , 4 , 6 } . A ∩ B = { 0 , 2 , 4 , 6 } . Next, find C ′ . C ′ . The complement of set C C is the set of elements in the universal set U U that are not in set C . C . C ′ = { 1 , 2 , 4 , 5 , 7 , 8 , 10 , 11 } . C ′ = { 1 , 2 , 4 , 5 , 7 , 8 , 10 , 11 } . Finally, find ( A ∩ B ) ∪ C ′ = { 0 , 2 , 4 , 6 } ∪ { 1 , 2 , 4 , 5 , 7 , 8 , 10 , 11 } = { 0 , 1 , 2 , 4 , 5 , 6 , 7 , 8 , 10 , 11 } . ( A ∩ B ) ∪ C ′ = { 0 , 2 , 4 , 6 } ∪ { 1 , 2 , 4 , 5 , 7 , 8 , 10 , 11 } = { 0 , 1 , 2 , 4 , 5 , 6 , 7 , 8 , 10 , 11 } .

Your Turn 1.37

Using the same sets from Example 1.37, perform the set operations indicated.

Notice that the answers to the Your Turn are the same as those in the Example. This is not a coincidence. The following equivalences hold true for sets:

• A ∩ ( B ∩ C ) = ( A ∩ B ) ∩ C A ∩ ( B ∩ C ) = ( A ∩ B ) ∩ C and A ∪ ( B ∪ C ) = ( A ∪ B ) ∪ C . A ∪ ( B ∪ C ) = ( A ∪ B ) ∪ C . These are the associative property for set intersection and set union.
• A ∩ B = B ∩ A A ∩ B = B ∩ A and A ∪ B = B ∪ A . A ∪ B = B ∪ A . These are the commutative property for set intersection and set union.
• A ∩ ( B ∪ C ) = ( A ∩ B ) ∪ ( A ∩ C ) A ∩ ( B ∪ C ) = ( A ∩ B ) ∪ ( A ∩ C ) and A ∪ ( B ∩ C ) = ( A ∪ B ) ∩ ( A ∪ C ) . A ∪ ( B ∩ C ) = ( A ∪ B ) ∩ ( A ∪ C ) . These are the distributive property for sets over union and intersection, respectively.

Proving Equality of Sets Using Venn Diagrams

To prove set equality using Venn diagrams, the strategy is to draw a Venn diagram to represent each side of the equality, then look at the resulting diagrams to see if the regions under consideration are identical.

Augustus De Morgan was an English mathematician known for his contributions to set theory and logic. De Morgan’s law for set complement over union states that ( A ∪ B ) ′ = A ′ ∩ B ′ ( A ∪ B ) ′ = A ′ ∩ B ′ . In the next example, we will use Venn diagrams to prove De Morgan’s law for set complement over union is true. But before we begin, let us confirm De Morgan’s law works for a specific example. While showing something is true for one specific example is not a proof, it will provide us with some reason to believe that it may be true for all cases.

Let U = { 1 , 2 , 3 , 4 , 5 , 6 , 7 } , U = { 1 , 2 , 3 , 4 , 5 , 6 , 7 } , A = { 2 , 3 , 4 } , A = { 2 , 3 , 4 } , and B = { 3 , 4 , 5 , 6 } . B = { 3 , 4 , 5 , 6 } . We will use these sets in the equation ( A ∪ B ) ′ = A ′ ∩ B ′ . ( A ∪ B ) ′ = A ′ ∩ B ′ . To begin, find the value of the set defined by each side of the equation.

Step 1: A ∪ B A ∪ B is the collection of all unique elements in set A A or set B B or both. A ∪ B = { 2 , 3 , 4 , 5 , 6 } . A ∪ B = { 2 , 3 , 4 , 5 , 6 } . The complement of A union B , ( A ∪ B ) ′ ( A ∪ B ) ′ , is the set of all elements in the universal set that are not in A ∪ B A ∪ B . So, the left side the equation ( A ∪ B ) ′ ( A ∪ B ) ′ is equal to the set { 1 , 7 } . { 1 , 7 } .

Step 2: The right side of the equation is A ′ ∩ B ′ . A ′ ∩ B ′ . A ′ A ′ is the set of all members of the universal set U U that are not in set A A . A ′ = { 1 , 5 , 6 , 7 } . A ′ = { 1 , 5 , 6 , 7 } . Similarly, B ′ = { 1 , 2 , 7 } . B ′ = { 1 , 2 , 7 } .

Step 3: Finally, A ′ ∩ B ′ A ′ ∩ B ′ is the set of all elements that are in both A ′ A ′ and B ′ . B ′ . The numbers 1 and 7 are common to both sets, therefore, A ′ ∩ B ′ = { 1 , 7 } . A ′ ∩ B ′ = { 1 , 7 } . Because, { 1 , 7 } = { 1 , 7 } { 1 , 7 } = { 1 , 7 } we have demonstrated that De Morgan’s law for set complement over union works for this particular example. The Venn diagram below depicts this relationship.

Example 1.38

Proving de morgan’s law for set complement over union using a venn diagram.

De Morgan’s Law for the complement of the union of two sets A A and B B states that: ( A ∪ B ) ′ = A ′ ∩ B ′ . ( A ∪ B ) ′ = A ′ ∩ B ′ . Use a Venn diagram to prove that De Morgan’s Law is true.

Step 1: First, draw a Venn diagram representing the left side of the equality. The regions of interest are shaded to highlight the sets of interest. A ∪ B A ∪ B is shaded on the left, and ( A ∪ B ) ′ ( A ∪ B ) ′ is shaded on the right.

Step 2: Next, draw a Venn diagram to represent the right side of the equation. A ′ A ′ is shaded and B ′ B ′ is shaded. Because A ′ A ′ and B ′ B ′ mix to form A ′ ∩ B ′ A ′ ∩ B ′ is also shaded.

Step 3: Verify the conclusion. Because the shaded region in the Venn diagram for ( A ∪ B ) ′ ( A ∪ B ) ′ matches the shaded region in the Venn diagram for A ′ ∩ B ' A ′ ∩ B ' , the two sides of the equation are equal, and the statement is true. This completes the proof that De Morgan’s law is valid.

Your Turn 1.38

Check your understanding, section 1.5 exercises.

A gamers club at Baily Middle School consisting of 25 members was surveyed to find out who played board games, card games, or video games. Use the results depicted in the Venn diagram below to answer the following exercises.

A blood drive at City Honors High School recently collected blood from 140 students, staff, and faculty. Use the results depicted in the Venn diagram below to answer the following exercises.

For the following exercises, perform the set operations as indicated on the following sets: U = { 20 , 21 , 22 , … , 29 } , A = { 21 , 24 , 27 } , B = { 20 , 22 , 24 , 28 } , and C = { 21 , 23 , 25 , 27 } .

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Sets - Problem Solving

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If \(A = \{ 1,2,3,4 \}, B = \{ 4,5,6,7 \},\) determine the following sets: (i) \(A \cap B\) (ii) \(A \cup B\) (iii) \(A \backslash B \) (i) By definition, \(\cap\) tells us that we want to find the common elements between the two sets. In this case, it is 4 only. Thus \(A \cap B = \{ 4 \} \). (ii) By definition, \(\cup\) tells us that we want to combine all the elements between the two sets. In this case, it is \(A\cup B = \{1,2,3,4,5,6,7 \} \). (iii) By definition, \( \backslash \) tells us that we want to look for elements in the former set in that doesn't appear in the latter set. So \(A \backslash B = \{1,2,3\} \). \(_\square\)

Consider the same example above. If the element \(4\) is removed from the set \(B\), solve for (i), (ii), (iii) as well. (i) Since there is no common elements in sets \(A\) and \(B\), then \(A \cap B = \phi \) or \(A \cap B = \{ \} \). (ii) Because the element \(4\) is no longer repeated, then \(A \cup B \) remains the same. (iii) Since \(A\) and \(B\) no longer share any common element, \(A\backslash B \) is simply equals to set \(A\), which is \(\{1,2,3,4 \} \). \(_\square\)

If \(P=\{2, 5, 6, 3, 7\}\) and \(Q=\{1, 2, 3, 8, 9, 10\},\) which of the following Venn diagrams represents the relationship between the two sets?

\[\large\color{darkred}{B=\{ \{ M,A,T,H,S \} \}}\]

Find the cardinal number of the set \(\color{darkred}{B}\).

Note: The cardinal number of a set is equal to the number of elements contained in the set.

Bonus question given with the picture.

Join the brilliant classes and enjoy the excellence. also checkout foundation assignment #2 for jee..

Consider the set \( \lbrace{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\rbrace}\).

For each of its subsets, let \( M \) be the greatest number. Find the last three digits of the sum of all the \( M \)'s.

Assume that \(0\) is the greatest number of the empty subset.

The number of subsets in set A is 192 more than the number of subsets in set B. How many elements are there in set A?

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Learning Objectives

In this chapter, you will learn to:

• Use set theory and Venn diagrams to solve counting problems.
• Use the Multiplication Axiom to solve counting problems.
• Use Permutations to solve counting problems.
• Use Combinations to solve counting problems.
• Use the Binomial Theorem to expand \((x+y)^n\)
• 7.1.1: Sets and Counting (Exercises)
• 7.2.1: Tree Diagrams and the Multiplication Axiom (Exercises)
• 7.3.1: Permutations (Exercises)
• 7.4.1: Circular Permutations and Permutations with Similar Elements (Exercises)
• 7.5.1: Combinations (Exercises)
• 7.6.1: Combinations- Involving Several Sets (Exercises)
• 7.7.1: Binomial Theorem (Exercises)
• 7.8: Chapter Review

Problem Solving: Introduction to Sets

Learn to read sets from a file and to sort and display them.

• Exercise 1: Loading set B
• Implementation of sorting set A
• Exercise 2: Sorting set B
• Exercise 3: Printing set B

In this lesson, we’ll work on sets. Sets are a well-defined collection of non-repetitive objects. In our case, we’ll only be looking at the sets of integers.

In this lesson, we’ll learn to:

• Read integer sets from a file
• Sort the integer sets
• Display the sets

So let’s start!

Reading sets from a file

Task: Load the two sets A and B from the file.

The file has the following format:

The two sizes (size of A and size of B) followed by set A and set B entries.

Here is a sample file:

As we know, a set cannot contain duplicate elements. Therefore, if a number is added twice, that duplicate should not be allowed to enter the set.

So we will have two sets, say A and B, both containing numbers without repetitions. The numbers can be either positive or negative.

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• Word Problems: Three sets

Venn Diagram Word Problem Worksheets - Three Sets

This page contains worksheets based on Venn diagram word problems, with Venn diagram containing three circles. The worksheets are broadly classified into two skills - Reading Venn diagram and drawing Venn diagram. The problems involving a universal set are also included. Printable Venn diagram word problem worksheets can be used to evaluate the analytical skills of the students of grade 6 through high school and help them organize the data. Access some of these worksheets for free!

Reading Venn Diagram - Type 1

These 6th grade pdf worksheets consist of Venn diagrams containing three sets with the elements that are illustrated with pictures. Interpret the Venn diagram and answer the word problems given below.

• Download the set

Reading Venn Diagram - Type 2

The elements of the sets are represented as symbols on the three circles of the Venn diagram. Count the symbols and write the answers in the space provided. Each worksheet contains two real-life scenarios.

Standard Word Problems - Without Universal Set

These PDF worksheets contain 3-circle Venn diagrams without universal set. Study the Venn diagram and answer the word problems.

Standard Word Problems - With Universal Set

This section features extensive collection of Venn diagram word problems with a universal set for grade 7 and grade 8 students. Real-life scenarios like doctor appointments, attractions of Niagara Falls and more are used.

Drawing Venn Diagram - Without Universal Set

Draw a venn diagram with three intersecting circles and fill in the data from the given descriptions. Also, ask 8th grade and high school students to answer questions based on union, intersection, and complement of three sets.

Draw Venn Diagram - With Universal Set

Six exclusive printable worksheets on Venn diagram word problems are included here. Draw three overlapping circles and fill in the regions using the information provided. Based on your findings, answer the word problems.

Related Worksheets

» Venn Diagram Word Problems - 2 sets

» Pictograph

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» Tally Marks

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Class 11 math (India)

Course: class 11 math (india)   >   unit 1, practical problems on union and intersection of sets (basic).

• Your answer should be
• an integer, like 6 ‍  
• a simplified proper fraction, like 3 / 5 ‍  
• a simplified improper fraction, like 7 / 4 ‍  
• a mixed number, like 1   3 / 4 ‍  
• an exact decimal, like 0.75 ‍  
• a multiple of pi, like 12   pi ‍   or 2 / 3   pi ‍  

Sets in mathematics, are simply a collection of distinct objects forming a group. A set can have any group of items, be it a collection of numbers, days of a week, types of vehicles, and so on. Every item in the set is called an element of the set. Curly brackets are used while writing a set. A very simple example of a set would be like this. Set A = {1, 2, 3, 4, 5}. In set theory, there are various notations to represent elements of a set. Sets are usually represented using a roster form or a set builder form. Let us discuss each of these terms in detail.

Sets Definition

In mathematics, a set is defined as a well-defined collection of objects. Sets are named and represented using capital letters. In the set theory, the elements that a set comprises can be any kind of thing: people, letters of the alphabet, numbers, shapes, variables, etc.

Sets in Maths Examples

Some standard sets in maths are:

• Set of natural numbers , ℕ = {1, 2, 3, ...}
• Set of whole numbers , W = {0, 1, 2, 3, ...}
• Set of integers , ℤ = {..., -3, -2, -1, 0, 1, 2, 3, ...}
• Set of rational numbers , ℚ = {p/q | q is an integer and q ≠ 0}
• Set of irrational numbers , ℚ' = {x | x is not rational}
• Set of real numbers , ℝ = ℚ ∪ ℚ'

All these are infinite sets. But there can be finite sets as well. For example, the collection of even natural numbers less than 10 can be represented in the form of a set, A = {2, 4, 6, 8}, which is a finite set.

Let us use this example to understand the basic terminology associated with sets in math.

Elements of a Set

The items present in a set are called either elements or members of a set. The elements of a set are enclosed in curly brackets separated by commas. To denote that an element is contained in a set, the symbol '∈' is used. In the above example, 2 ∈ A. If an element is not a member of a set, then it is denoted using the symbol '∉'. For example, 3 ∉ A.

Cardinal Number of a Set

The cardinal number, cardinality , or order of a set denotes the total number of elements in the set. For natural even numbers less than 10, n(A) = 4. Sets are defined as a collection of unique elements. One important condition to define a set is that all the elements of a set should be related to each other and share a common property. For example, if we define a set with the elements as the names of months in a year, then we can say that all the elements of the set are the months of the year.

Representation of Sets in Set Theory

There are different set notations used for the representation of sets in set theory. They differ in the way in which the elements are listed. The three set notations used for representing sets are:

• Semantic form
• Roster form
• Set builder form

Let us understand each of these forms with an example.

Semantic Form

Semantic notation describes a statement to show what are the elements of a set. For example, a set of the first five odd numbers .

Roster Form

The most common form used to represent sets is the roster notation in which the elements of the sets are enclosed in curly brackets separated by commas. For example, Set B = {2,4,6,8,10}, which is the collection of the first five even numbers. In a roster form, the order of the elements of the set does not matter, for example, the set of the first five even numbers can also be defined as {2,6,8,10,4}. Also, if there is an endless list of elements in a set, then they are defined using a series of dots at the end of the last element. For example, infinite sets are represented as, X = {1, 2, 3, 4, 5 ...}, where X is the set of natural numbers. To sum up the notation of the roster form, please take a look at the examples below.

Finite Roster Notation of Sets : Set A = {1, 2, 3, 4, 5} (The first five natural numbers)

Infinite Roster Notation of Sets : Set B = {5, 10, 15, 20 ....} (The multiples of 5 )

Set Builder Form

The set builder notation has a certain rule or a statement that specifically describes the common feature of all the elements of a set. The set builder form uses a vertical bar in its representation, with a text describing the character of the elements of the set. For example, A = { k | k is an even number, k ≤ 20}. The statement says, all the elements of set A are even numbers that are less than or equal to 20. Sometimes a ":" is used in the place of the "|".

Visual Representation of Sets Using Venn Diagram

Venn Diagram is a pictorial representation of sets, with each set represented as a circle. The elements of a set are present inside the circles . Sometimes a rectangle encloses the circles, which represents the universal set . The Venn diagram represents how the given sets are related to each other.

Sets Symbols

Set symbols are used to define the elements of a given set. The following table shows the set theory symbols and their meaning.

Types of Sets

There are different types of sets in set theory. Some of these are singleton, finite, infinite, empty, etc.

Singleton Sets

A set that has only one element is called a singleton set or also called a unit set. Example, Set A = { k | k is an integer between 3 and 5} which is A = {4}.

Finite Sets

As the name implies, a set with a finite or countable number of elements is called a finite set . Example, Set B = {k | k is a prime number less than 20}, which is B = {2,3,5,7,11,13,17,19}

Infinite Sets

A set with an infinite number of elements is called an infinite set. Example: Set C = { Multiples of 3 }.

Empty or Null Sets

A set that does not contain any element is called an empty set or a null set. An empty set is denoted using the symbol '∅'. It is read as ' phi '. Example: Set X = { }.

If two sets have the same elements in them, then they are called equal sets . Example: A = {1,2,3} and B = {1,2,3}. Here, set A and set B are equal sets. This can be represented as A = B.

Unequal Sets

If two sets have at least one different element, then they are unequal sets. Example: A = {1,2,3} and B = {2,3,4}. Here, set A and set B are unequal sets. This can be represented as A ≠ B.

Equivalent Sets

Two sets are said to be equivalent sets when they have the same number of elements, though the elements are different. Example: A = {1,2,3,4} and B = {a,b,c,d}. Here, set A and set B are equivalent sets since n(A) = n(B)

Overlapping Sets

Two sets are said to be overlapping if at least one element from set A is present in set B. Example: A = {2,4,6} B = {4,8,10}. Here, element 4 is present in set A as well as in set B. Therefore, A and B are overlapping sets.

Disjoint Sets

Two sets are disjoint if there are no common elements in both sets. Example: A = {1,2,3,4} B = {5,6,7,8}. Here, set A and set B are disjoint sets.

Subset and Superset

For two sets A and B, if every element in set A is present in set B, then set A is a subset of set B(A ⊆ B) and in this case, B is the superset of set A(B ⊇ A). Example: Consider the sets A = {1,2,3} and B = {1,2,3,4,5,6}. Here:

• A ⊆ B, since all the elements in set A are present in set B.
• B ⊇ A denotes that set B is the superset of set A.

Universal Set

A universal set is the collection of all the elements regarding a particular subject. The universal set is denoted by the letter 'U'. Example: Let U = {The list of all road transport vehicles}. Here, a set of cars is a subset for this universal set, the set of cycles, trains are all subsets of this universal set.

Power set is the set of all subsets that a set could contain. Example: Set A = {1,2,3}. Power set of A is = {∅, {1}, {2}, {3}, {1,2}, {2,3}, {1,3}, {1,2,3}}.

Operations on Sets

Some important operations on sets in set theory include union, intersection, difference, the complement of a set, and the cartesian product of a set. A brief explanation of set operations is as follows.

Union of Sets

Union of sets, which is denoted as A U B, lists the elements in set A and set B or the elements in both set A and set B. For example, {1, 3} ∪ {1, 4} = {1, 3, 4}

Intersection of Sets

The intersection of sets which is denoted by A ∩ B lists the elements that are common to both set A and set B. For example, {1, 2} ∩ {2, 4} = {2}

Set Difference

Set difference which is denoted by A - B, lists the elements in set A that are not present in set B. For example, A = {2, 3, 4} and B = {4, 5, 6}. A - B = {2, 3}.

Set Complement

Set complement which is denoted by A', is the set of all elements in the universal set that are not present in set A. In other words, A' is denoted as U - A, which is the difference in the elements of the universal set and set A.

Cartesian Product of Sets

The cartesian product of two sets which is denoted by A × B, is the product of two non-empty sets, wherein ordered pairs of elements are obtained. For example, {1, 3} × {1, 3} = {(1, 1), (1, 3), (3, 1), (3, 3)}.

In the above figure, the shaded portions in "blue" show the set that they are labelled with.

Sets Formulas in Set Theory

Sets find their application in the field of algebra , statistics , and probability . There are some important set theory formulas in set theory as listed below.

For any two overlapping sets A and B,

• n(A U B) = n(A) + n(B) - n(A ∩ B)
• n (A ∩ B) = n(A) + n(B) - n(A U B)
• n(A) = n(A U B) + n(A ∩ B) - n(B)
• n(B) = n(A U B) + n(A ∩ B) - n(A)
• n(A - B) = n(A U B) - n(B)
• n(A - B) = n(A) - n(A ∩ B)

For any two sets A and B that are disjoint,

• n(A U B) = n(A) + n(B)
• n(A - B) = n(A)

Properties of Sets

Similar to numbers, sets also have properties like associative property, commutative property, and so on. There are six important properties of sets . Given, three sets A, B, and C, the properties for these sets are as follows.

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Sets Examples

Example 1: Find the elements of the sets represented as follows and write the cardinal number of each set. a) Set A is the first 8 multiples of 7 b) Set B = {a,e,i,o,u} c) Set C = {x | x are even numbers between 20 and 40}

a) Set A = {7,14,21,28,35,42,49,56}. These are the first 8 multiples of 7 .

Since there are 8 elements in the set, cardinal number n (A) = 8

b) Set B = {a,e,i,o,u}. There are five elements in the set,

Therefore, the cardinal number of set B, n(B) = 5. c) Set C = {22,24,26,28,30,32,34,36,38}. These are the even numbers between 20 and 40, which make up the elements of the set C.

Therefore, the cardinal number of set C, n(C) = 9.

Answer: (a) 8 (b) 5 (c) 9

Example 2: If Set A = {a,b,c}, Set B = {a,b,c,p,q,r}, U = {a,b,c,d,p,q,r,s}, find the following using sets formulas, a) A U B b) A ∩ B c) A' d) Is A ⊆ B? (Here 'U' is the universal set).

a) A U B = writing the elements of A and B together in one set by removing duplicates = {a,b,c,p,q,r}

b) A ∩ B = writing common elements of A and B in a set = {a,b,c}

c) A' = writing elements of U that are NOT present in A = {d,p,q,r,s}

d) A ⊆ B, (Set A is a subset of set B) since all the elements in set A are present in set B.

Answer: (a) {a,b,c,p,q,r} (b) {a,b,c} (c) {d,p,q,r,s} (d) Yes

Example 3: Express the given set in set-builder form: A = {2, 4, 6, 8, 10, 12, 14}

Solution: Given: A = {2, 4, 6, 8, 10, 12, 14}

Using sets notations, we can represent the given set A in set-builder form as,

A = {x | x is an even natural number less than 15}

Answer: A = {x | x is an even natural number less than 15}

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Practice Questions on Sets

Faqs on sets, what is set in math.

Sets are a collection of distinct elements, which are enclosed in curly brackets, separated by commas. The list of items in a set is called the elements of a set. Examples are a collection of fruits, a collection of pictures. Sets are represented by the symbol { }. i.e., the elements of the set are written inside these brackets. Example: Set A = {a,b,c,d}. Here, a,b,c, and d are the elements of set A.

What are Different Sets Notations to Represent Sets?

Sets can be represented in three ways. Representing sets means a way of listing the elements of the set. They are as follows.

• Semantic Notation: The elements of a set are represented by a single statement. For example, Set A is the number of days in a week.
• Roster Notation: This form of representation of sets uses curly brackets to list the elements of the set. For example, Set A = {10,12,14,16,18}}
• Set Builder Notation: A set builder form represents the elements of a set by a common rule or a property. For example, {x | x is a prime number less than 20}

What are the Types of Sets?

Sets differ from each depending upon elements present in them. Based on this, we have the following types of sets . They are singleton sets, finite and infinite sets, empty or null sets, equal sets, unequal sets, equivalent sets, overlapping sets, disjoint sets, subsets, supersets, power sets, and universal sets.

What are the Properties of Sets in Set-Theory?

Different properties associated with sets in math are,

• Commutative Property: A U B = B U A and A ∩ B = B ∩ A
• Associative Property: (A ∩ B) ∩ C = A ∩ (B ∩ C) and (A U B) U C = A U (B U C)
• Distributive Property: A U (B ∩ C) = (A U B) ∩ (A U C) and A ∩ (B U C) = (A ∩ B) U (A ∩ C)
• Identity Property: A U ∅ = A and A ∩ U = A
• Complement Property: A U A' = U
• Idempotent Property: A ∩ A = A and A U A = A

What is the Union of Sets?

The union of two sets A and B are the elements from both set A and B, or both combined together. It is denoted using the symbol 'U'. For example, if set A = {1,2,3} and set B = {4,5,6}, then A U B = {1,2,3,4,5,6}. A U B is read as 'A union B'.

What is the Intersection of Sets?

The intersection of two sets A and B are the elements that are common to both set A and B. It is denoted using the symbol '∩'. For example, if set A = {1,2,3} and set B = {3,4,5}, then A ∩ B = {3}. A ∩ B is read as 'A intersection B'.

What are Subsets and Supersets?

If every element in a set A is present in set B, then set B is the superset of set A and set A is a subset of set B. Example: A = {1,4,5} B = {1,2,3,4,5,6}, here since all elements of set A are present in set B ⇒ A ⊆ B and B ⊇ A.

What are Universal Sets?

A universal set, denoted by the letter 'U', is the collection of all the elements in regard to a particular subject. Example: Let U = {All types of cycles}. Here, a set of cycles of a specific company is a subset of this universal set.

What Does Sets Class 11 Contain?

The sets in class 11 is an important chapter that deals with various components of set theory. It starts with definition of sets, and extends to types of sets, properties of sets, set operations, etc. It also has some real-life applications related to sets. To solve more applications related to sets class 11, click here . ☛Also Check:

• NCERT Solutions Class 11 Maths Chapter 1 Ex 1.1
• NCERT Solutions Class 11 Maths Chapter 1 Ex 1.2
• NCERT Solutions Class 11 Maths Chapter 1 Ex 1.3
• NCERT Solutions Class 11 Maths Chapter 1 Ex 1.4
• NCERT Solutions Class 11 Maths Chapter 1 Ex 1.5
• NCERT Solutions Class 11 Maths Chapter 1 Ex 1.6
• NCERT Solutions Class 11 Maths Chapter 1 Miscellaneous Exercise

What is Complement in Sets?

The complement of a set which is denoted by A', is the set of all elements in the universal set that are not present in set A. In other words, A' is denoted as U - A, which is the difference in the elements of the universal set and set A.

What is Cartesian Product in Sets?

Cartesian product of two sets, denoted by A×B, is the product of two non-empty sets, wherein ordered pairs of elements are obtained. For example, if A = {1,2} and B = {3,4}, then A×B = {(1,3), (1,4), (2,3), (2,4)}.

What is the Use of Venn Diagram in Set Theory?

Venn Diagram is a pictorial representation of the relationship between two or more sets. Circles are used to represent sets. Each circle represents a set. A rectangle that encloses the circles represents the universal set.

Browse Course Material

Course info, instructors.

• Prof. Arthur Mattuck
• Prof. Haynes Miller
• Dr. Jeremy Orloff
• Dr. John Lewis

Departments

• Mathematics

As Taught In

• Differential Equations
• Linear Algebra

Learning Resource Types

Problem sets with solutions.

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• Math Formulas
• Sets Formulas

Set Formulas

A collection of objects is called a Set.

Formulas of Sets

These are the basic set of formulas from the set theory.

If there are two sets P and Q,

• n(P U Q) represents the number of elements present in one of the sets P or Q.
• n(P  ⋂ Q) represents the number of elements present in both the sets P & Q.
• n(P U Q) = n(P) + (n(Q) – n (P ⋂ Q)

For three sets P, Q, and R,

• \(\begin{array}{l}n(P U Q U R) = n(P) + n(Q) + n(R) – n(P\bigcap Q) – n(Q\bigcap R) – n(R\bigcap P) + n(P\bigcap Q\bigcap R)\end{array} \)

Examples of Sets Formulas

Example 1: In a class, there are 100 students, 35 like drawing and 45 like music. 10 like both. Find out how many of them like either of them or neither of them?

Number of drawing students, n(d) = 35

Number of music students, n(m) = 45

Number of students who like both, n(d∩m) = 10

Number of students who like either of them,

n(dᴜm) = n(d) + n(m) – n(d∩m)

→ 45+35-10 = 70

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