Sets (Copy)
Practice Questions – 1.2 Sets
Question 1
Write down the meaning of each notation clearly:
(a) n(A)
(b) A′
(c) A ∪ B
(d) A ∩ B
Question 2
Let A = {2, 4, 6, 8}, B = {4, 8, 10, 12}.
(a) Find A ∪ B.
(b) Find A ∩ B.
(c) Find n(A ∪ B).
Question 3
The universal set U = {1, 2, 3, 4, 5, 6, 7, 8, 9}.
Let P = {even numbers in U}, Q = {prime numbers in U}.
(a) Write P and Q in full.
(b) Find P ∩ Q.
(c) Find P′.
Question 4
A survey of 50 students found that:
- 28 study Mathematics (M)
- 20 study Physics (P)
- 12 study both.
(a) Represent this information on a Venn diagram.
(b) How many students study Mathematics only?
(c) How many study Physics only?
(d) How many study neither subject?
Question 5
In a school:
- 60 students play football (F)
- 45 students play cricket (C)
- 25 students play both sports.
If there are 100 students in total:
(a) Draw a Venn diagram.
(b) Find the number of students who play only football.
(c) Find the number of students who play only cricket.
(d) Find the number who play neither sport.
Question 6
Let A = {x: x is a factor of 24}, B = {x: x is a factor of 36}.
(a) Write A and B in full.
(b) Find A ∩ B.
(c) Find A ∪ B.
(d) State whether A ⊆ B or not.
Question 7
If A = {1, 2, 3, 4}, B = {2, 4, 6}, C = {1, 3, 5, 7}:
(a) Find A ∩ B
(b) Find (A ∩ B) ∪ C
(c) Find A ∩ (B ∪ C)
Question 8
In a group of 80 students:
- 35 like tea (T)
- 30 like coffee (C)
- 20 like both tea and coffee.
(a) Draw a Venn diagram.
(b) How many students like tea only?
(c) How many like coffee only?
(d) How many like neither?
Question 9
Let A = {x: 1 ≤ x ≤ 12, x is a multiple of 3}, B = {x: 1 ≤ x ≤ 12, x is a factor of 12}.
(a) Write A and B in full.
(b) Find A ∩ B.
(c) Find A′ given the universal set U = {1, 2, …, 12}.
Question 10
True or false? Explain.
(a) If A ⊆ B, then A ∩ B = A.
(b) A ∩ B = ∅ means A and B have no elements in common.
(c) The complement of the universal set is ∅.
(d) If n(A) = 0, then A = ∅.
Answer key and explanations — 1.2 Sets
1.
(a) n(A) = number of elements in set A
(b) A′ = complement of A (all elements in the universal set U but not in A)
(c) A ∪ B = union of A and B (all elements in A or B or both)
(d) A ∩ B = intersection of A and B (all elements in both A and B)
Explanation: These are the standard notations in set theory.
2.
A = {2, 4, 6, 8}, B = {4, 8, 10, 12}
(a) A ∪ B = {2, 4, 6, 8, 10, 12}
(b) A ∩ B = {4, 8}
(c) n(A ∪ B) = 6
Explanation: union collects all without repetition, intersection collects common elements.
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course
3.
U = {1, 2, 3, 4, 5, 6, 7, 8, 9}
P = {2, 4, 6, 8}, Q = {2, 3, 5, 7}
(a) P = {2, 4, 6, 8}, Q = {2, 3, 5, 7}
(b) P ∩ Q = {2}
(c) P′ = {1, 3, 5, 7, 9}
Explanation: complement is within universal set; only even primes overlap here.
4.
- Total = 50, n(M) = 28, n(P) = 20, n(M ∩ P) = 12
(a) Venn diagram: 12 in overlap, 16 in M only, 8 in P only, 14 outside.
(b) 28 − 12 = 16
(c) 20 − 12 = 8
(d) 50 − (16+12+8) = 14
Explanation: subtract overlaps when filling Venn diagram.
5.
- Total = 100, n(F) = 60, n(C) = 45, n(F ∩ C) = 25
(a) Venn: overlap 25, football only 35, cricket only 20, neither 20
(b) Football only = 60 − 25 = 35
(c) Cricket only = 45 − 25 = 20
(d) Neither = 100 − (35+25+20) = 20
Explanation: Use total count to calculate “neither.”
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course
6.
Factors of 24: A = {1, 2, 3, 4, 6, 8, 12, 24}
Factors of 36: B = {1, 2, 3, 4, 6, 9, 12, 18, 36}
(a) Written above
(b) A ∩ B = {1, 2, 3, 4, 6, 12}
(c) A ∪ B = {1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36}
(d) A is not a subset of B, because A has 8 and 24 which are not in B.
Explanation: subset means every element of A is also in B.
7.
A = {1, 2, 3, 4}, B = {2, 4, 6}, C = {1, 3, 5, 7}
(a) A ∩ B = {2, 4}
(b) (A ∩ B) ∪ C = {1, 2, 3, 4, 5, 7}
(c) B ∪ C = {1, 2, 3, 4, 5, 6, 7}, then A ∩ (B ∪ C) = {1, 2, 3, 4} = A
Explanation: union first, then intersect carefully.
8.
Total = 80, T = 35, C = 30, T ∩ C = 20
(a) Venn: 20 overlap, 15 in tea only, 10 in coffee only, 35 in neither
(b) Tea only = 35 − 20 = 15
(c) Coffee only = 30 − 20 = 10
(d) Neither = 80 − (15+20+10) = 35
Explanation: always subtract overlap when counting “only.”
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course
9.
U = {1,2,…,12}
A = {3, 6, 9, 12} (multiples of 3)
B = {1, 2, 3, 4, 6, 12} (factors of 12)
(a) Listed above
(b) A ∩ B = {3, 6, 12}
(c) A′ = U A = {1, 2, 4, 5, 7, 8, 10, 11}
Explanation: complement means everything in U except those in A.
10.
(a) True: if A ⊆ B then all of A lies inside B, so their intersection is A.
(b) True: ∩ being empty means no shared elements.
(c) True: universal set includes everything, so its complement is empty.
(d) True: if n(A) = 0, A has no elements, so A = ∅.
Explanation: follow definitions strictly.
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course