problem solving involving 2 sets

Reset password New user? Sign up

Existing user? Log in

Sets - Problem Solving

Already have an account? Log in here.

This wiki is incomplete.

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?

Problem Loading...

Note Loading...

Set Loading...

Set Operations

Set operations is a concept similar to fundamental operations on numbers. Sets in math deal with a finite collection of objects, be it numbers, alphabets, or any real-world objects. Sometimes a necessity arises wherein we need to establish the relationship between two or more sets. There comes the concept of set operations.

There are four main set operations which include set union, set intersection, set complement, and set difference. In this article, we will learn the various set operations, notations of representing sets, how to operate on sets, and their usage in real life.

What are Set Operations?

A set is defined as a collection of objects. Each object inside a set is called an 'Element'. A set can be represented in three forms. They are statement form, roster form , and set builder notation . Set operations are the operations that are applied on two or more sets to develop a relationship between them. There are four main kinds of set operations which are as follows.

• Union of sets
• Intersection of sets
• Complement of a set
• Difference between sets/Relative Complement

Before we move on to discuss the various set operations, let us recall the concept of Venn diagrams as it is important in understanding the operations on sets. A Venn diagram is a logical diagram that shows the possible relationship between different finite sets. The Venn diagram can be represented as follows.

Basic Set Operations

Now that we know the concept of a sets and Venn diagram, let us discuss each set operation one by one in detail. The various set operations are:

Union of Sets

For two given sets A and B, A∪B (read as A union B) is the set of distinct elements that belong to set A and set B or both. The number of elements in A ∪ B is given by n(A∪B) = n(A) + n(B) − n(A∩B), where n(X) is the number of elements in set X. To understand this set operation of the union of sets better, let us consider an example: If A = {1, 2, 3, 4} and B = {4, 5, 6, 7}, then the union of A and B is given by A ∪ B = {1, 2, 3, 4, 5, 6, 7}.

Intersection of Sets

For two given sets A and B, A∩B (read as A intersection B) is the set of common elements that belong to set A and B. The number of elements in A∩B is given by n(A∩B) = n(A)+n(B)−n(A∪B), where n(X) is the number of elements in set X. To understand this set operation of the intersection of sets better, let us consider an example: If A = {1, 2, 3, 4} and B = {3, 4, 5, 7}, then the intersection of A and B is given by A ∩ B = {3, 4}.

Set Difference

The set operation difference between sets implies subtracting the elements from a set which is similar to the concept of the difference between numbers. The difference between sets A and set B denoted as A − B lists all the elements that are in set A but not in set B. To understand this set operation of set difference better, let us consider an example: If A = {1, 2, 3, 4} and B = {3, 4, 5, 7}, then the difference between sets A and B is given by A - B = {1, 2}.

Complement of Sets

The complement of a set A denoted as A′ or A c (read as A complement) is defined as the set of all the elements in the given universal set(U) that are not present in set A. To understand this set operation of complement of sets better, let us consider an example: If U = {1, 2, 3, 4, 5, 6, 7, 8, 9} and A = {1, 2, 3, 4}, then the complement of set A is given by A' = {5, 6, 7, 8, 9}.

The above image shows various set operations with the help of Venn diagrams. When the elements of one set B completely lie in the other set A, then B is said to be a proper subset of A. When two sets have no elements in common, then they are said to be disjoint sets . Now, let us explore the properties of the set operations.

Properties of Set Operations

The properties of set operations are similar to the properties of fundamental operations on numbers. The important properties on set operations are stated below:

• Commutative Law - For any two given sets A and B, the commutative property is defined as, A ∪ B = B ∪ A This means that the set operation of union of two sets is commutative. A ∩ B = B ∩ A This means that the set operation of intersection of two sets is commutative.
• Associative Law - For any three given sets A, B and C the associative property is defined as, (A ∪ B) ∪ C = A ∪ (B ∪ C) This means the set operation of union of sets is associative. (A ∩ B) ∩ C = A ∩ (B ∩ C) This means the set operation of intersection of sets is associative.
• De-Morgan's Law - The De Morgan's law states that for any two sets A and B, we have (A ∪ B)' = A' ∩ B' and (A ∩ B)' = A' ∪ B'

Important Notes on Set Operations

• Set operation formula for union of sets is n(A∪B) = n(A) + n(B) − n(A∩B) and set operation formula for intersection of sets is n(A∩B) = n(A)+n(B)−n(A∪B).
• The union of any set with the universal set gives the universal set and the intersection of any set A with the universal set gives the set A.
• Union, intersection, difference, and complement are the various operations on sets.
• The complement of a universal set is an empty set U′ = ϕ. The complement of an empty set is a universal set ϕ′ = U.

Related Topics on Set Operations

• Finite and Infinite Sets

Examples of Set Operations

Example 1: In a school, every student plays either football or soccer or both. It was found that 200 students played football, 150 students played soccer and 100 students played both. Find how many students were there in the school using the set operation formula.

Solution: Let us represent the number of students who played football as n(F) and the number of students who played soccer as n(S). We have n(F) = 200, n(S) = 150 and n(F ∩ S) = 100. We know that,

n(F∪S) = n(F) + n(S) − n(F∩S)

Therefore, n(F∪S)=(200+150)−100

n(F∪S) = 350 − 100 = 250

Answer: Hence the total number of students in the school is 250.

Example 2: If A = {a, b, c, d, e}, B = {a, e, i, o, u}, U = {a, b, c, d, e, f, g, h, i, j, k, l, o, u}. Perform the following operations on sets and find the solutions.

a) A ∪ B b) A ∩ B c) A′ d) A - B

Solution: a) A ∪ B = {a, b, c, d, e, i, o, u}

b) A ∩ B = {a, e}

c) A' = {f, g, h, i, j, k, l, o, u}

d) A - B = {b, c, d}

go to slide go to slide

Book a Free Trial Class

Practice Questions on Set Operations

Faqs on set operations, what are set operations in set theory.

Set operations are the operations that are applied on two or more sets to develop a relationship between them. There are four main kinds of set operations.

What are the Different Set Operations?

There are four main kinds of set operations which are:

How Do We Use Set Operations in Real Life?

A set is a collection of elements. Some real-life examples of sets are a list of all the states in a country, a list of all shapes in geometry , list of all whole numbers from 1 to 100. We can determine the common regions using the intersection set operation.

How do you Solve Set Operations Problems?

To solve set operation problems we use a Venn diagram to represent the relationship between the sets and apply the set operations formula for union, intersection, difference, or complement of a set.

Which of the Set Operations are Commutative and not Commutative?

Union and Intersection of sets are set operations that are commutative whereas the set difference is not commutative.

What are the Set Operations Symbols?

There are different symbols used for different set operations are referred as set notations . For the union of sets, we use ' ∪ ', for the intersection of sets, we use '∩', for the difference of sets, we use ' - ', and for the complement of a set A, we write it as A' or A c .

How do you Find the Difference Between the Two Sets?

For any two sets A and B, the difference A - B lists all the elements in set A that are not in set B.

How do you Find the Complement of a Set?

Given the universal set 'U' and set A, the complement of a set A is defined as the set of all elements in the universal set that are not present in set A.

What are the Union and Intersection operations of Sets?

For any two sets A and B, the union is defined as the combination of elements in both set A and B. Intersection of sets gives the common elements in set A and set B.

Stack Exchange Network

Stack Exchange network consists of 183 Q&A communities including Stack Overflow , the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Q&A for work

Connect and share knowledge within a single location that is structured and easy to search.

Solving equations involving two sets

Out of curiosity, how would one go about solving an equation involving two sets? For example,

$$ \{1, 2, 3\} = \{a + b + c, a + b - c, a - b + c, a - b - c\} $$

An intuitive solution to this is $ \{a = 2, b = 0.5, c = 0.5\} $ , but is there a specific process?

• elementary-set-theory
• 1 $\begingroup$ Set's dont have a strict order. So you have to go through all possibilities, for example $1=a+b+c$, $1=a+b-c$ and $1=a-b-c$. I don't if two sets with different cardinality can be set equal. Don't think so. $\endgroup$ –  vitamin d Commented Feb 20, 2021 at 19:15
• 1 $\begingroup$ $\{1,2,3\}$ has three elements and $\{a+b+c,a+b−c,a−b+c,a−b−c\}$ has $4$ so the second must have repeated elements. $\endgroup$ –  fleablood Commented Feb 20, 2021 at 19:55

2 Answers 2

Notice the set on the left has three distinct elements, but the set on the right has four representations so two of those represetations are of the same number.

So we have six options.

$a+b+c = a+b-c$ and $c = 0$ and we have the values $a+b$ and $a-b$ . But that's only two different values (or fewer) and we have exactly $3$ so that's impossible.

$a+b + c = a-b+c$ and $b =0$ and have the values $a+c$ and $a-c$ and that's the same problem.

$a+b+c = a-b-c$ and $b+c= 0$ and $b=-c$ and we have the values $a, a+2b, a-2b$ . We'll get back to this.

$a+b-c = a-b + c$ and $b-c=0$ and $b=c$ and we have the values $a+2b; a; a-2b$ . That will have the same solutions as above.

$a+b-c=a-b-c$ and $b=0$ and we have the same problem we had earlier.

Or $a-b+c = a-b-c$ adn $c=0$ and we have the same problem from the very begining.

So either way we have $a,a+2b, a-2b = 1,2,3$ .

$a\pm 2b, a , a\pm 2b$ are in arithmetic progression so we must have either $a-2b < a < a+2b$ and $a-2b = 1, a=2, a+2b=3$ or $a+2b < a < a-2b$ and $a+2b = 1, a=2, a-2b =3$

So we must have $a=2$ and $b =\pm 0.5$ and $c = \pm 0.5$ (so there are four sets of solutions.)

There isn't really any way to do this in general.

In the right set, two elements are equal. The only possibilities are $$a+b+c=a-b-c$$ or $$a+b-c=a-b+c$$ which gives $$b=\pm c$$

The equation becomes $$\{a-2b,a,a+2b\}=\{1,2,3\}$$

The sum gives $ 3a=6 $ and $ a=2$ .

You must log in to answer this question.

• Featured on Meta
• We spent a sprint addressing your requests — here’s how it went
• Upcoming initiatives on Stack Overflow and across the Stack Exchange network...

Hot Network Questions

• 8x8 grid with no unmarked L-pentomino
• Mathematical Induction over two numbers
• Did Joe Biden refer to himself as a black woman?
• Are there countries where only voters affected by a given policy get to vote on it?
• Why bother with planetary battlefields?
• Rolling median of all K-length ranges
• Can Black Lotus alone enable the Turn 1 Channel-Fireball combo?
• Using "Delight" Without a Preposition
• What enforcement exists for medical informed consent?
• Finding the Zeckendorf Representation of a Positive Integer (BIO 2023 Q1)
• Does the damage from Thunderwave occur before or after the target is moved
• Keyboard Ping Pong
• Why do jet aircraft need chocks when they have parking brakes?
• Intelligence vs Wisdom in D&D
• How can one count how many pixels a GIF image has via command line?
• Is it possible to have multiple versions of MacOS on the same laptop at the same time?
• How can I search File Explorer for files only (i.e. exclude folders) in Windows 10?
• How is 11:22 four minutes slow if it's actually 11:29?
• Can I set QGIS to summarize icons above a certain scale and display only the number of icons in an area?
• Can I prettify a XML string using Apex?
• Alternatives to iterrow loops in python pandas dataframes
• Why seperating a sphere gives different shades?
• Is it an option for the ls utility specified in POSIX.1-2017?
• Why does black have a higher win rate in Exchange French?

• Math Article

Sets Questions

Sets questions with solutions are given here for students to make them understand the concept easily. Practising these problems will help to go through the concept of sets theory . It is an important chapter for Class 11 students, hence we have given the questions based on the NCERT curriculum, with respect to the CBSE syllabus.

A brief introduction for each sub-topic of sets is also provided so that students can solve the questions quickly. Some of the basic definitions related to this concept are given below. Also, read Sets For Class 11 , here.

Sets : A collection of well-defined objects. It is denoted by capital letters.

Example: A = {1, 2, 3, 4, 5..}.

Here, set A is a collection of all natural numbers.

Roster form of sets: All elements are written in curly braces { }, separated by commas.

Example: R = {1, 3, 7, 21, 2, 6, 14, 42}.

Set Builder Form: The elements of the set represent a common property.

Example, R = {x : x is a vowel in English alphabet}

First, let us see some questions based on the representation of sets.

Questions on Sets with Solutions

1. Write the solution set of the equation x 2 – 4=0 in roster form.

Solution: x 2 – 4 = x 2 – 2 2 = (x – 2) (x + 2)

Thus, A = {-2, 2}

2. Write the set A = {1, 4, 9, 16, 25, . . . } in set-builder form.

Solution: If we see the pattern here, the numbers are squares of natural numbers, such as:

A = {x : x is the square of a natural number}

Or we can write;

A = {x : x = n 2 , where n ∈ N}

3. Write an example of a finite and infinite set in set builder form.

Finite set, A = {x : x ∈ N and (x – 1) (x – 2) = 0}

Infinite Set, B = {x : x ∈ N and x is prime}

4. Write an example of equal sets.

Solution: Let there be two sets A and B

A is the set of letters in “ALLOY”

B is the set of letters in “LOYAL”

A = {A,L, O,Y}

B = {L,O,Y,A}

Therefore, in both sets, the elements are the same.

5. Write the subsets of {1,2,3}.

Solution: Let A = {1, 2, 3}

The subsets of A are: φ, {1}, {2}, {3}, {1, 2}, {2, 3}, {1, 3}, {1, 2, 3}

6. Write {x: x ∈ R, 3 ≤ x ≤ 4} as an interval.

7. Write the interval (6, 12) in set builder form.

Let A be the interval (6, 12).

The interval (6, 12) in set builder form is

A = {x: x ∈ R, 6 < x < 12}

8. If set A = {1, 3, 5}, B = {2, 4, 6} and C = {0, 2, 4, 6, 8}. Then write the universal set for all three sets.

Solution: If U is the universal set for sets A, B and C, then: U = Elements of set A + Elements of set B + Elements of set C

U = {0, 1, 2, 3, 4, 5, 6, 8}

Also, read: Sets Subset And Superset

9. If A = { 2, 4, 6, 8} and B = { 6, 8, 10, 12}. Find A ∪ B.

Solution: A ∪ B = { 2, 4, 6, 8, 10, 12}

10. If A = { 2, 4, 6, 8} and B = { 6, 8, 10, 12}. Find A ∩ B.

Solution: A ∩ B = { 6, 8 }

Also, see : Operation On Sets Intersection Of Sets And Difference Of Two Sets

11. If A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {5, 6, 7, 8}. Find A ∪ B ∪ C.

Solution: A ∪ B ∪ C = {1, 2, 3, 4, 5, 6, 7, 8}

12. If A = {3, 5, 7, 9, 11}, B = {7, 9, 11, 13}, C = {11, 13, 15}. Find A ∩ (B ∪ C).

As we know, A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

= {7, 9, 11} ∪ {11}

= {7, 9, 11}

13. If A = { 1, 2, 3, 4, 5, 6}, B = { 2, 4, 6, 8 }. Find A – B and B – A.

Solution: A – B = { 1, 3, 5 }

B – A = { 8 }

Clearly, A – B ≠ B – A

14. If U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and A = {1, 3, 5, 7, 9}. Find A′.

Solution: A′ = { 2, 4, 6, 8, 10 }

Video Lesson on What are Sets

Practice Questions

Solve the following questions on sets:

• Check whether the given sets are equal sets: A = {1, 2, 3, 4} and B = {2, 4, 1, 3}.
• Write the subsets for the set A = {1, 3, 5, 7}
• Write the set A = {1, 2, 3, 4, 5, …} in set-builder form.
• If A = {1, 3, 5, 7, 9, 11} and B = {1, 2, 3, 13}, the find A – B and B – A.
• Find A ∪ (B ∪ C), if A = {1, 3, 5}, B = {2, 4, 6} and C = {1, 5, 7}.

Keep visiting BYJU’S – The Learning App and download the app today to get all Maths-related concepts and video lessons to learn the concepts quickly.

Leave a Comment Cancel reply

Your Mobile number and Email id will not be published. Required fields are marked *

Request OTP on Voice Call

Post My Comment

A servey was conducted among 300 students . It was found that 125 students like to play cricket . 145 students like to play football . 90 students like to play tennis . 32 students like to play exactly two games out of the three games. How many students like to play exactly one games .?

n(C) = 125, n(F) = 145, n(T) = 90 n(C ⋃ F ⋃ T) = 125 + 145 + 90 – [n(C ⋂ F) + n(F ⋂ T) + n(T ⋂ C)] + n(c ⋂ F ⋂ T) 300 = 360 – [n(C ⋂ F) + n(F ⋂ T) + n(T ⋂ C)] + n(c ⋂ F ⋂ T) n(C ⋂ F) + n(F ⋂ T) + n(T ⋂ C) = 60 + n(c ⋂ F ⋂ T)….(i) Also, given that, n(C ⋂ F) + n(F ⋂ T) + n(T ⋂ C) – 3[n(c ⋂ F ⋂ T)] = 32 n(C ⋂ F) + n(F ⋂ T) + n(T ⋂ C) = 32 + 3[n(c ⋂ F ⋂ T)]….(ii) From (i) and (ii), 60 + n(c ⋂ F ⋂ T) = 32 + 3[n(c ⋂ F ⋂ T)] 60 – 32 = 3[n(c ⋂ F ⋂ T)] – n(c ⋂ F ⋂ T) 2[n(c ⋂ F ⋂ T)] = 28 n(c ⋂ F ⋂ T) = 14 Number of students like to play exactly one game = n(C) + n(F) + n(T) – 2[n(C ⋂ F) + n(F ⋂ T) + n(T ⋂ C)] + 3[n(c ⋂ F ⋂ T)] = 125 + 145 + 90 – 2[32 + 3 × 14] + 3 × 14 = 360 – 148 + 42 = 254

Register with BYJU'S & Download Free PDFs

Register with byju's & watch live videos.

Browse Course Material

Course info, instructors.

• Dr. Jeremy Orloff
• Dr. Jennifer French Kamrin

Departments

• Mathematics

As Taught In

• Discrete Mathematics
• Probability and Statistics

Learning Resource Types

Introduction to probability and statistics, problem sets with solutions.

You are leaving MIT OpenCourseWare

If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

To log in and use all the features of Khan Academy, please enable JavaScript in your browser.

Course: 3rd grade   >   Unit 8

• Setting up 2-step word problems
• Represent 2-step word problems with equations
• 2-step word problem: truffles

2-step word problem: running

• 2-step word problem: theater
• 2-step word problems

Want to join the conversation?

• Upvote Button navigates to signup page
• Downvote Button navigates to signup page
• Flag Button navigates to signup page

Video transcript