Union and Intersection of Sets Practice Questions

Union and the intersection of sets questions are fundamental in understanding set theory, a crucial concept in mathematics. Mastering these operations allows you to solve problems related to various mathematical disciplines, including probability, statistics, and algebra.

In this article, we will learn about some of the fundamental operations of sets with the help of Union and the intersection of sets questions. These unions and the intersection of sets of questions will help you to understand the concepts and apply them to solve problems in various fields such as computer science, statistics, and logic.

Important Formulas for Union and Intersection of Sets

Some of the common formulas and properties related to Union and Intersection of Sets are:

Union and Intersection of Sets Practice Questions - Solved

1. Set A contains A = {1, 2, 3} and set B contains B = {3, 4, 5}. Find A ∪ B and ? ∩ ?.

Set A = {1, 2, 3}

Set B = {3, 4, 5}

A ∪ B = {1, 2, 3, 4, 5}

? ∩ ? = {3}

2. In a class of 30 students, 20 students play football and 15 students play basketball. How many students play either football or basketball? How many students play both?

Total no of students = 30

student playing football = n(f) = 20

student playing basketball = n(b) = 15

To find the students who play both, use this formula

n(f ∪ b) = n(f) + n(b) - n(f ∩ b)

30 = 20 + 15 - n(f ∩ b)

n(f ∩ b) = 35 - 30

n(f ∩ b) = 5

So 5 students can play both football and basketball.

3. Represent the set A = {x : x is an even number less than 10} and B = {x : x is a multiple of 3}. Calculate A ∪ B and A ∩ B.

Set A = {x : x is an even number less than 10}

Set A = {2, 4, 6, 8}

Set B = {x : x is a multiple of 3}

Set B = {3, 6, 9}

A ∪ B = {2, 3, 4, 6, 8, 9}

A ∩ B = {6}

4. Draw a Venn diagram representing two sets A and B. Label the regions to represent A ∪ B and A ∩ B.

5. Verify the commutative property of union and intersection for two sets A and B by using specific sets.

Set A = {1, 2, 3}

Set B = {3, 4, 5}

Commutative property of Union of sets

A ∪ B = B ∪ A

Now,

A ∪ B = {1, 2, 3, 4, 5}

B ∪ A = {1, 2, 3, 4, 5}

So, A ∪ B = B ∪ A

Commutative property of Intersection of sets

A ∩ B = B ∩ A

Now,

A ∩ B = {3}

B ∩ A = {3}

So, A ∩ B = B ∩ A.

6. If U = {1, 2, 3, 4, 5, 6} is the universal set, and A = {2, 4, 6} and B = {3, 4, 5}, find A ∪ B and A ∩ B.

Given sets,

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

A = {2, 4, 6}

B = {3, 4, 5}

A ∪ B = {2, 3, 4, 5, 6}

A ∩ B = {4}

7. Let U = {1, 2, 3, 4, 5, 6}, A = {1, 2, 3}, and B = {4, 5, 6}. Calculate A ∩ B′, where ?′ is the complement of set B in U.

Given sets,

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

A = {1, 2, 3}

B = {4, 5, 6}

B' = {1, 2, 3}

A ∩ B′ = {1, 2, 3}

8. In a group of 40 students, 25 like pizza, and 20 like burgers. If 15 students like both pizza and burgers, what is the probability that a randomly selected student likes either pizza or burgers?

Total no of students = 40

P(p) = 25/40 = 5/8 (probability of liking pizza)

P(b) = 20/40 = 1/2 (probability of liking burgers)

P(p ∩ b) = 15/40 = 3/8 (probability of liking both pizza and burgers)

P( p ∪ b) = P(p) + P(b) − P(p ∩ b)

⇒ P( p ∪ b) = 5/8 + 1/2 - 3/8

⇒ P( p ∪ b) = 6/8

⇒ P( p ∪ b) = 3/4

So, the probability that a randomly selected student likes either pizza or burgers is 3/4.

9. We have a group of 50 students, and 30 students like pizza (A), and 25 students like burgers (B). Additionally, 15 students like both pizza and burgers (A ∩ B). Using De Morgan's laws, we can find the complement of A ∪ B.

Total number of students = 50

Students liking pizza = n(A) = 30

Students liking burgers = n(B) = 25

Students like both pizza and burgers (A ∩ B) = 15

A′(students who do not like pizza): 50−30 = 20 students.

B′(students who do not like burgers): 50−25 = 25 students.

(A ∪ B)′ = A′ ∩ B′ = 20

So, students who do not like either pizza or burgers is 20

10. Consider two shopping lists; List A contains fruits (apples, oranges, bananas), and List B contains vegetables (carrots, tomatoes, spinach).

Create a new list representing items that are either fruits or vegetables, and another list representing items that are both fruits and vegetables.

Given set of items are

List A = {apples, oranges, bananas}

List B = {carrots, tomatoes, spinach}

Now, list of either fruits or vegetables is list C i.e.,

List C = {apples, oranges, bananas, carrots, tomatoes, spinach}

Now, list of items which are both fruits and vegetables is A ∩ B = List D i.e.,

List D = ∅

Union and Intersection of Sets Practice Questions -Unsolved

Q1: Set A contains A = {2, 4, 6} and set B contains B = {3, 4, 5}. Find A ∪ B and A ∩ B.

Q2: In a class of 35 students, 22 students play football and 18 students play basketball. How many students play either football or basketball? How many students play both?

Q3: Represent the set A = {x : x is a multiple of 4 and less than 10} and B = {x : x is an odd number). Calculate A ∪ B and A ∩ B.

Q4: Draw a Venn diagram representing two sets A and B. Label the regions to represent A ∪ B and A ∩ B.

Q5: Verify the associative property of union and intersection for two sets A and B where A = {1, 3, 5} and B = {2, 3, 4}.

Q6: If U = {1, 2, 3, 4, 5, 6, 7} is the universal set, and A = {2, 4, 6} and B = {3, 5, 7}, find A ∪ B and A ∩ B.

Q7: Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A = {1, 2, 3, 4}, and B = {4, 5, 6, 7}. Calculate A ∩ B', where B′ is the complement of set B in U.

Q8: In a group of 50 students, 30 like pizza, and 25 like burgers. If 20 students like both pizza and burgers, what is the probability that a randomly selected student likes either pizza or burgers?

Q9: We have a group of 60 students, and 35 students like pizza (A), and 28 students like burgers (B). Additionally, 18 students like both pizza and burgers (A ∩ B). Using De Morgan's laws, we can find the complement of A ∪ B.

Q10: Consider two shopping lists: List A contains fruits (apples, oranges, bananas), and List B contains vegetables (carrots, tomatoes, spinach, potatoes). Create a new list representing items that are either fruits or vegetables, and another list representing items that are both fruits and vegetables.

Union and Intersection of Sets Practice Questions -Unsolved Answers

Ans 1: A ∪ B = {2,3,4,5,6} and A ∩ B = {4}

Ans 2: 5 students play both and 30 students play either one

Ans 3: A = {4,8} and B = {1,3,5,7....} , A ∪ B = {1,3,4,5,7,8,9,11.....} and A ∩ B = { } = ∅

Ans 4:

Ans 5: A ∪ (B ∪ C) = (A ∪ B) ∪ C and A ∩ (B ∩ C) = (A ∩ B) ∩ C]

Ans 6: A ∪ B = {2,3,4,5,6,7} and A ∩ B = { } = ∅

Ans 7: A ∩ B' = {1,2,3}

Ans 8: 7/10

Ans 9: (A ∪ B)' = 15

Ans 10: A ∪ B = {Apples, Oranges, Bananas, Carrots, Tomatoes, Spinach, Potatoes} and A ∩ B = { } = ∅

Also Read,

• What is Set Theory?
• Set Operations
• Representation of a Set

Conclusion

Understanding the union and intersection of sets is crucial for solving problems in various fields such as computer science, statistics, and logic. Through the practice problems and examples provided, you can see how these concepts are applied to real-world scenarios. By mastering the operations of union and intersection, as well as their properties and formulas, you can enhance your problem-solving skills and build a solid foundation in set theory.