11th NCERT Sets Exercise 1.5 Questions 7
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Question (1)

Let U ={1, 2, 3; 4, 5, 6, 7, 8, 9}, A = {1, 2, 3, 4}, B = {2, 4, 6, 8} and C = {3, 4, 5, 6}. Find
(i) A'

Solution

A' = { x : x ∈ U, x ∉ A}
A' = { 5, 6, 7, 8, 9}
(ii) B'

Solution

B' = { x : x ∈ U, x ∉ B}
B' = { 1, 3, 5, 7, 9}
(iii)( A∪C)'

Solution

( A ∪ C) = { 1, 2, 3, 4, 5, 6}
(A ∪C)' = { x : x ∈ U, x ∉( A ∪ C)
( A ∪ C)' = { 7, 8, 9}
(iv) ( A ∪ B)'

Solution

( A ∪ B) = { 1, 2, 3, 4, 6, 8}
(A ∪B)' = { x : x ∈ U, x ∉( A ∪ B)}
( A ∪ B)' = { 5, 7, 9}
(v) (A')"

Solution

A' = { x : x ∈ U, x ∉ A}
A' = { 5, 6, 7, 8, 9}
(A')' = { x ∈ U, x ∉ A'}
= {1, 2, 3, 4} = A
(vi) (B- C)'

Solution

B ∩ C = { 4,6}
B - C = { x : x ∈ B, x ∉ C}
= { 2, 8}
(B - C)' = { x : x ∈ U, x ∉( B - C)}
= { 1, 3, 4, 5, 6, 7, 9}

Question (2)

If U = {a, b, c, d, e, f, g, h}, find the complements of the following sets:
(i) A = {a, b, c}

Solution

A' = { x : x ∈ U, x ∉ A}
= { d, e ,f, g, h}
(ii) B = {d, e, f, g}

Solution

B' = { x : x ∈ U, x ∉ B}
= { a, b, c, h}
(iii) C = {a, c, e, g}

Solution

C' = { x : x ∈ U, x ∉ C}
= { b, d, f. h}
(iv) D = {f, g, h, a}

Solution

D' = { x : x ∈ U, x ∉ D}
= { b. c. d. e}

Question (3)

Taking the set of natural numbers as the universal set, write down the complements of the following sets:
(i) {x: x is an even natural number}

Solution

A = { 2, 4, 6, ....}
A' = { x : x ∈ U, x ∉ A}
= { 1, 3, 5, 7, ....}
= set of odd natural numbers.
(ii) {x: x is an odd natural number}

Solution

A = { 1, 3, 5, 7, ....}
A' = { x : x ∈ U, x ∉ A}
= { 2, 4, 6, 8, ....}
= { x: x is even natural numbers}
(iii) {x: x is a positive multiple of 3}

Solution

A = { 3, 6, 9, 12,.....}
A' = { x : x ∈ U, x ∉ A}
= {1, 2, 4, 5, 7, 8, .....}
= {x: x ∈ N, x is not multiple of 3}
(iv) {x: x is a prime number}

Solution

A = { 2, 3, 5,7, 11, 13, ...}
A' = { x : x ∈ U, x ∉ A}
= { 1, 4, 6, 8, 9, 10, 12,.....}
= {Set of composite numbers} ∪ {1}
(v) {x: x is a natural number divisible by 3 and 5}

Solution

A = { 15, 30, 45, ....}
A' = { x : x ∈ U, x ∉ A}
= { 1,2,3,...14,16,17,...29, 31,...}
= {x: x ∈ N, x is not divisible by 3 and 5}
(vi) {x: x is a perfect square}

Solution

A = { 1, 4, 9, 16, ....}
A' = { x : x ∈ U, x ∉ A}
= { x : x ∈ N, x is not perfect square}
(vii) {x: x is perfect cube}

Solution

A = { 1, 8, 27, 64, ..}
A' = { x : x ∈ U, x ∉ A}
= {x: x∈ N, x is not perfect cube}
(viii) {x: x + 5 = 8}

Solution

A = {3}
A' = { x : x ∈ U, x ∉ A}
= N - {3}
(ix) {x: 2x + 5 = 9}

Solution

A = {2}
A' = { x : x ∈ U, x ∉ A}
= N - {2}
(x) {x: x ≥ 7}

Solution

A = {7, 8, 9, 10, ....}
A' = { x : x ∈ U, x ∉ A}
= {1, 2, 3, 4, 5, 6}
(xi) {x: x ∈ N and 2x + 1 > 10}

Solution

A = { 5, 6, 7, 8, ....}
A' = { x : x ∈ U, x ∉ A}
= {1, 2. 3, 4}

Question (4)

If U = {1, 2, 3, 4, 5,6,7,8, 9}, A = {2, 4, 6, 8} and B = {2, 3, 5, 7}. Verify that
(i) ( A ∪ B)' = A' ∩ B'

Solution

A = {2, 4, 6, 8} and B = {2, 3, 5, 7}
A ∪ B = { 2, 3, 4, 5, 6, 7, 8}
LHS = ( A ∪ B)'
= {1, 9}
A' = { 1, 3, 5, 7, 9} and B' = { 1, 4, 6, 8, 9}
RHS = A' ∩ B'
= {1, 9}
= LHS
∴ ( A ∪ B)' = A' ∩ B'
(ii) ( A ∩ B)' = A' ∪ B'

Solution

A = {2, 4, 6, 8} and B = {2, 3, 5, 7}
A ∩ B = { 2}
LHS = ( A ∩ B)'
= { 1, 3, 4, 5, 6, 7, 8, 9}
A' = { 1, 3, 5, 7, 9} and B' = { 1, 4, 6, 8, 9}
RHS = A' ∪ B'
= { 1, 3, 4, 5, 6, 7, 8, 9}
= LHS
∴ ( A ∩ B)' = A' ∪ B'

Question (5)

Draw appropriate Venn diagram for each of the following:
(i) (A ∪B)'

Solution

(ii) A'∩B'

Solution

(iii)(A∩B)'

Solution

(iv) A'∪B'

Solution

Question (6)

Let U be the set of all triangles in a plane. If A is the set of all triangles with at least one angle different from 60°, what is A'?

Solution

U = set of all triangles.
A = { all triangles with atleast one angle is different than 600}
A' = { x : x ∈ U, x ∉ A}
A' = { a triangle with all angle 600}
= set of equilateral triangles.

Question (7)

Fill in the blanks to make each of the following a true statement:

Solution

(i) A ∪ A' =   U
(ii) Φ′ ∩ A = …   A
(iii)A ∩ A' =   φ
(iv) U' ∩ A =   Φ