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}
(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 perfect cube}
(viii) {x: x + 5 = 8}
(ix) {x : 2x + 5 = 9}
(x) {x : x ≥ 7}
(xi) {x : x ∈ N and 2x + 1 > 10}

Solution:

It is given that the universal set is,

U = N = Set of natural numbers

We know that the complement of a set A is denoted by A' and it is equal to U - A, where U is the universal set. Thus,

(i) {x : x is an even natural number}´ = {x : x is an odd natural number}

(ii) {x : x is an odd natural number}´ = {x : x is an even natural number}

(iii) {x : x is a positive multiple of 3}´ = {x : x ∈ N and x is not a multiple of 3}

(iv) {x : x is a prime number}´ = { x : x is a positive composite number or x = 1}

(v) {x : x is a natural number divisible by 3 and 5}´ = {x : x is a natural number that is not divisible by 3 or not divisible by 5}

(vi) {x : x is a perfect square}´ = {x : x ∈ N and x is not a perfect square}

(vii) { x : x is a perfect cube}´ = {x : x Î N and x is not a perfect cube}

(viii) We have {x: x + 5 = 8} = { x: x = 3}.

Thus, its complement is

{x: x + 5 = 8}' = {x : x ∈ N and x ≠ 3}

(ix) We have {x : 2x + 5 = 9} = { x : 2x = 4} = {x : x = 2}.

Thus, its complement is

{x : 2x + 5 = 9}´ = {x : x ∈ N and x ≠ 2}

(x) {x : x ≥ 7}´ = {x : x ∈ N and x < 7}

(xi) We have {x : x ∈ N and 2x + 1 > 10} = {x : x ∈ N and 2x > 9} = {x : x ∈ N and x > 9/2}.

Thus, its complement is

{x : x ∈ N and 2x + 1 > 10}´ = {x : x ∈ N and x ≤ 9/2}

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NCERT Solutions Class 11 Maths Chapter 1 Exercise 1.5 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} (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 }

Summary:

Taking the set of natural numbers as the universal set, we are asked to write down the complements of the given sets. We found that:
(i) {x : x is an even natural number}´ = {x : x is an odd natural number}
(ii) {x : x is an odd natural number}´ = {x : x is an even natural number}
(iii) {x : x is a positive multiple of 3}´ = {x : x ∈ N and x is not a multiple of 3}
(iv) {x : x is a prime number}´ ={ x : x is a positive composite number or x = 1}
(v) {x : x is a natural number divisible by 3 and 5}´ = {x : x is a natural number that is not divisible by 3 or not divisible by 5}
(vi) {x : x is a perfect square}´ = {x : x ∈ N and x is not a perfect square}
(vii) { x : x is a perfect cube}´ = {x : x Î N and x is not a perfect cube}
(viii) {x: x + 5 = 8}´ = {x : x ∈ N and x ≠ 3}
(ix) {x : 2x + 5 = 9}´ = {x : x ∈ N and x ≠ 2}
(x) {x : x ≥ 7}´ = {x : x ∈ N and x < 7}
(xi) {x : x ∈ N and 2x + 1 > 10}´ = {x : x ∈ N and x ≤ 9/2}