📚 Exercise 1.5 - Sets
Complement of a Set - Complete Solutions
1
Let U = {1,2,3,4,5,6,7,8,9}, A = {1,2,3,4}, B = {2,4,6,8}, C = {3,4,5,6}. Find
(i) A′ (ii) B′ (iii) (A ∪ C)′
(iv) (A ∪ B)′ (v) (A′)′ (vi) (B – C)′
(iv) (A ∪ B)′ (v) (A′)′ (vi) (B – C)′
💡 Click to View Solution
1 Find A′
A′ = U – A = {1,2,3,4,5,6,7,8,9} – {1,2,3,4}
= {5, 6, 7, 8, 9}
= {5, 6, 7, 8, 9}
2 Find B′
B′ = U – B = {1,2,3,4,5,6,7,8,9} – {2,4,6,8}
= {1, 3, 5, 7, 9}
= {1, 3, 5, 7, 9}
3 Find (A ∪ C)′
A ∪ C = {1,2,3,4} ∪ {3,4,5,6} = {1,2,3,4,5,6}
(A ∪ C)′ = U – (A ∪ C)
= {1,2,3,4,5,6,7,8,9} – {1,2,3,4,5,6}
= {7, 8, 9}
(A ∪ C)′ = U – (A ∪ C)
= {1,2,3,4,5,6,7,8,9} – {1,2,3,4,5,6}
= {7, 8, 9}
4 Find (A ∪ B)′
A ∪ B = {1,2,3,4} ∪ {2,4,6,8} = {1,2,3,4,6,8}
(A ∪ B)′ = U – (A ∪ B)
= {1,2,3,4,5,6,7,8,9} – {1,2,3,4,6,8}
= {5, 7, 9}
(A ∪ B)′ = U – (A ∪ B)
= {1,2,3,4,5,6,7,8,9} – {1,2,3,4,6,8}
= {5, 7, 9}
5 Find (A′)′
A′ = {5,6,7,8,9}
(A′)′ = U – A′ = {1,2,3,4,5,6,7,8,9} – {5,6,7,8,9}
= {1, 2, 3, 4} = A
Key: The complement of complement equals the original set
(A′)′ = U – A′ = {1,2,3,4,5,6,7,8,9} – {5,6,7,8,9}
= {1, 2, 3, 4} = A
Key: The complement of complement equals the original set
6 Find (B – C)′
B – C = {2,4,6,8} – {3,4,5,6} = {2,8}
(B – C)′ = U – (B – C)
= {1,2,3,4,5,6,7,8,9} – {2,8}
= {1, 3, 4, 5, 6, 7, 9}
(B – C)′ = U – (B – C)
= {1,2,3,4,5,6,7,8,9} – {2,8}
= {1, 3, 4, 5, 6, 7, 9}
2
If U = {a,b,c,d,e,f,g,h}, find the complements of the following sets
(i) A = {a,b,c} (ii) B = {d,e,f,g}
(iii) C = {a,c,e,g} (iv) D = {f,g,h,a}
(iii) C = {a,c,e,g} (iv) D = {f,g,h,a}
💡 Click to View Solution
1 Find A′
A′ = U – A = {a,b,c,d,e,f,g,h} – {a,b,c}
= {d, e, f, g, h}
= {d, e, f, g, h}
2 Find B′
B′ = U – B = {a,b,c,d,e,f,g,h} – {d,e,f,g}
= {a, b, c, h}
= {a, b, c, h}
3 Find C′
C′ = U – C = {a,b,c,d,e,f,g,h} – {a,c,e,g}
= {b, d, f, h}
= {b, d, f, h}
4 Find D′
D′ = U – D = {a,b,c,d,e,f,g,h} – {f,g,h,a}
= {b, c, d, e}
= {b, c, d, e}
3
Taking natural numbers as universal set, find complements of
(i) {x : x is an even natural number}
(ii) {x : x is an odd natural number}
(iii) {x : x is a positive multiple of 3}
(iv) {x : x is a prime number}
(v) {x : x is a natural number divisible by 3 and 5}
(vi) {x : x is a perfect square}
(vii) {x : x is a perfect cube}
(viii) {x : x + 5 = 8}
(ix) {x : 2x + 5 = 9}
(x) {x : x ≥ 7}
(xi) {x : x ∈ N and 2x + 1 > 10}
(ii) {x : x is an odd natural number}
(iii) {x : x is a positive multiple of 3}
(iv) {x : x is a prime number}
(v) {x : x is a natural number divisible by 3 and 5}
(vi) {x : x is a perfect square}
(vii) {x : x is a perfect cube}
(viii) {x : x + 5 = 8}
(ix) {x : 2x + 5 = 9}
(x) {x : x ≥ 7}
(xi) {x : x ∈ N and 2x + 1 > 10}
💡 Click to View Solution
1 Complements
(i) {x : x is an odd natural number}
(ii) {x : x is an even natural number}
(iii) {x : x ∈ N and x is not a multiple of 3}
(iv) {x : x is composite or x = 1}
(Composite: natural number > 1 that has divisors other than 1 and itself)
(Composite: natural number > 1 that has divisors other than 1 and itself)
(v) {x : x ∈ N and x is not divisible by 3 or not divisible by 5}
(vi) {x : x ∈ N and x is not a perfect square}
(vii) {x : x ∈ N and x is not a perfect cube}
(viii) {x : x ∈ N and x ≠ 3}
(Since x + 5 = 8 ⇒ x = 3)
(Since x + 5 = 8 ⇒ x = 3)
(ix) {x : x ∈ N and x ≠ 2}
(Since 2x + 5 = 9 ⇒ x = 2)
(Since 2x + 5 = 9 ⇒ x = 2)
(x) {x : x ∈ N and x < 7} = {1, 2, 3, 4, 5, 6}
(xi) {x : x ∈ N and x ≤ 4.5} = {1, 2, 3, 4}
(Since 2x + 1 > 10 ⇒ x > 4.5)
(Since 2x + 1 > 10 ⇒ x > 4.5)
4
If U = {1,2,3,4,5,6,7,8,9}, A = {2,4,6,8}, B = {2,3,5,7}. Verify
(i) (A ∪ B)′ = A′ ∩ B′ (De Morgan's Law)
(ii) (A ∩ B)′ = A′ ∪ B′ (De Morgan's Law)
(ii) (A ∩ B)′ = A′ ∪ B′ (De Morgan's Law)
💡 Click to View Solution
1 Verify (A ∪ B)′ = A′ ∩ B′
Left Side:
A ∪ B = {2,4,6,8} ∪ {2,3,5,7} = {2,3,4,5,6,7,8}
(A ∪ B)′ = U – (A ∪ B) = {1,2,3,4,5,6,7,8,9} – {2,3,4,5,6,7,8}
(A ∪ B)′ = {1, 9}
Right Side:
A′ = U – A = {1,3,5,7,9}
B′ = U – B = {1,4,6,8,9}
A′ ∩ B′ = {1,3,5,7,9} ∩ {1,4,6,8,9} = {1, 9}
∴ (A ∪ B)′ = A′ ∩ B′ ✓ VERIFIED
A ∪ B = {2,4,6,8} ∪ {2,3,5,7} = {2,3,4,5,6,7,8}
(A ∪ B)′ = U – (A ∪ B) = {1,2,3,4,5,6,7,8,9} – {2,3,4,5,6,7,8}
(A ∪ B)′ = {1, 9}
Right Side:
A′ = U – A = {1,3,5,7,9}
B′ = U – B = {1,4,6,8,9}
A′ ∩ B′ = {1,3,5,7,9} ∩ {1,4,6,8,9} = {1, 9}
∴ (A ∪ B)′ = A′ ∩ B′ ✓ VERIFIED
2 Verify (A ∩ B)′ = A′ ∪ B′
Left Side:
A ∩ B = {2,4,6,8} ∩ {2,3,5,7} = {2}
(A ∩ B)′ = U – (A ∩ B) = {1,2,3,4,5,6,7,8,9} – {2}
(A ∩ B)′ = {1,3,4,5,6,7,8,9}
Right Side:
A′ = {1,3,5,7,9}
B′ = {1,4,6,8,9}
A′ ∪ B′ = {1,3,5,7,9} ∪ {1,4,6,8,9} = {1,3,4,5,6,7,8,9}
∴ (A ∩ B)′ = A′ ∪ B′ ✓ VERIFIED
A ∩ B = {2,4,6,8} ∩ {2,3,5,7} = {2}
(A ∩ B)′ = U – (A ∩ B) = {1,2,3,4,5,6,7,8,9} – {2}
(A ∩ B)′ = {1,3,4,5,6,7,8,9}
Right Side:
A′ = {1,3,5,7,9}
B′ = {1,4,6,8,9}
A′ ∪ B′ = {1,3,5,7,9} ∪ {1,4,6,8,9} = {1,3,4,5,6,7,8,9}
∴ (A ∩ B)′ = A′ ∪ B′ ✓ VERIFIED
💡 Key Concept: De Morgan's Laws state that the complement of a union equals the intersection of complements, and the complement of an intersection equals the union of complements.
5
Fill in the blanks to make true statements
(i) A ∪ A′ = ...
(ii) φ′ ∩ A = ...
(iii) A ∩ A′ = ...
(iv) U′ ∩ A = ...
(ii) φ′ ∩ A = ...
(iii) A ∩ A′ = ...
(iv) U′ ∩ A = ...
💡 Click to View Solution
1 Properties of Complements
(i) A ∪ A′ = U
Every element is either in A or in A′
Every element is either in A or in A′
(ii) φ′ ∩ A = U ∩ A = A
The complement of empty set is the universal set
The complement of empty set is the universal set
(iii) A ∩ A′ = φ
No element can be both in A and in A′
No element can be both in A and in A′
(iv) U′ ∩ A = φ ∩ A = φ
The complement of universal set is empty set
The complement of universal set is empty set
Summary:
(i) U (ii) A (iii) φ (iv) φ
(i) U (ii) A (iii) φ (iv) φ
6
Let U be the set of all triangles. If A is the set of triangles with at least one angle ≠ 60°, what is A′?
💡 Click to View Solution
1 Understanding A′
A′ = U – A
= Set of all triangles with NO angle different from 60°
= Set of all triangles with EACH angle = 60°
A′ = Set of all equilateral triangles
(In an equilateral triangle, all three angles are 60°)
= Set of all triangles with NO angle different from 60°
= Set of all triangles with EACH angle = 60°
A′ = Set of all equilateral triangles
(In an equilateral triangle, all three angles are 60°)
