Taking the set of natural numbers as the ...

NCERT Class XI Mathematics - Sets - Solutions

Question : 45 of 73

Marks: +1, -0

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}

Solution:

Here U = {x : x ∈ N}

(i) Let A = {x : x is an even natural number}

A′ = U – A = {x : x ∈ N} – {x : x is an even natural number} = {x : x is an odd natural number}

(ii) Let A = {x : x is an odd natural number} A′ = U – A = {x : x ∈ N} – {x : x is an odd natural number} = {x : x is an even natural number}

(iii) Let A = {x : x is a positive multiple of 3}

∴ A′ = U – A = {x : x ∈ N} – {x : x is a positive multiple of 3} = {x : x ∈ N, x is not a multiple of 3}

(iv) Let A = {x : x is a prime number}

A′ = U – A = {x : x ∈ N} – {x : x is a prime number} = {x : x ∈ N, x is not a prime number} or, {x : x is positive composite number and x = 1}.

(v) Let A = {x : x is a natural number divisible by 3 and 5}

A′ = U – A = {x : x ∈ N} – {x : x is a natural number divisible by 3 and 5} = {x : x ∈ N}

– {x : x is a natural number divisible by 15} = {x : x ∈ N, x is not divisible by 15}

(vi) Let A = {x : x is a perfect square} A′ = U – A = {x : x ∈ N} – {x : x is a perfect square} = {x : x ∈ N, x is not a perfect square}

(vii) Let A = {x : x is a perfect cube} A′ = U – A = {x : x ∈ N} – {x : x is a perfect cube} = {x : x ∈ N, x is not a perfect cube}

(viii) Let A = {x : x + 5 = 8} = {3} A′ = U – A = {x : x ∈ N} – {3} = {x : x ∈ N, x ≠ 3}

(ix) Let A = {x : 2x + 5 = 9} = {2} A′ = U – A = {x : x ∈ N} – {2} = {x : x ∈ N, x ≠ 2}

(x) Let A = {x : x ≥ 7} = {7, 8, 9, 10, ......} A′ = U – A = {x : x ∈ N} – {7, 8, 9, 10,......} = {1, 2, 3, 4, 5, 6} = {x : x ∈ N and x < 7}

(xi) Let A = {x : x ∈ N and 2x + 1 > 10}

= {x : x ∊ N and x > 9/2}

∴ A′ = U – A = {x : x ∈ N and x ≤ 9/2}

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