Exercise 1.5

 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′       (ii) B′      (iii) (A ∪ C)′

 (iv) (A ∪ B)′          (v) (A′)′    (vi) (B – C)′

Answer:

i) A'=U-A={ 5, 6, 7, 8, 9}

ii) B'= U-B= {1, 3, 5, 7, 9 }

iii) (A ∪ C)={1, 2, 3, 4, 5, 6} now (AUC)'= U- (AUC)= {7, 8, 9 }

iv) (A ∪ B)={1, 2, 3, 4, 6, 8} now (AUB)'=U-(AUB)={5, 7,9 }

v) (A′)′  = A 

vi) B-C={ 2,8} now (B-C)' =U-(B-C)= {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}

Answer:

i) A' = U-A ={d, e, f, g, h }

ii) B'= U-B= { a, b, c, h}

iii) C'= U-C= { b, d, f, h}

iv) D'=U-D= {b, c, d, e}

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 perfect cube}

(viii) {x: x + 5 = 8}

(ix) {x: 2x + 5 = 9}

(x) {x: x ≥ 7}

(xi) {x: x ∈ N and 2x + 1 > 10}

Answer:

U = N: Set of natural numbers

(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 and 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 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 ∈ 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}

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'       ii)  (A∩ B)' = A'∪ B' 

Answer:

i)  (A∪ B)' = { 2, 3, 4, 5, 6, 7, 8}'= {1,9}

    A' ∩ B' =   {1, 3, 5, 7, 9}  ∩ {1, 4, 6, 8, 9}= { 1, 9}  HENCE PROVED 

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

   A'∪ B' = {1, 3, 5, 7, 9} ∪ {1, 4, 6, 8, 9} = { 1,3, 4, 5, 6, 7, 8, 9}  HENCE PROVED 

5. Draw appropriate Venn diagram for each of the following:

i)  (A∪ B)'  ii) A' ∩ B'  iii) (A∩ B)'  iv) A'∪ B'

Answer:

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' ?

Answer:

A' is the set of all equilateral triangles.

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

(i) A ∪ A' =.....       U

(ii) Φ′ ∩ A = …    U ∩ A = A

(iii) A∩ A' =......   Φ

(iv) U′ ∩ A =...    Φ ∩ A = Φ