**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) (AUC)’**

**(iv) (AUB)’**

**(v)(A’)’**

**(vi)(B-C)’**

Solution:

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}

(i) A’ = {5, 6, 7, 8, 9}

(ii)B’ = {1, 3, 5, 7, 9}

(iii) AUC = { 1, 2, 3, 4, 5, 6}

Thus, (AUC)’ = {7, 8, 9}

(iv) AUB = {1, 2, 3, 4, 6, 8}

Thus, (AUB)’ = {5, 7, 9}

(v) (A’)’ = A = {1, 2, 3, 4}

(vi) B – C = {2, 8}

Thus, (B – C)’ = {1, 3, 4, 5, 6, 7, 9}

**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}**

Solution:

U = {a, b, c, d, e, f, g, h}

(i) A = {a, b, c}

A’ = {d, e, f, g, h}

(ii) B = {d, e, f, g}

B’ = {a, b, c, h}

(iii) C = {a, c, e, g}

C’ = {b, d, f, h}

(iv) D = {f, g, h, a}

D’ = {b, c, d, e}

**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:

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}

**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) (AUB)’ = A’ ∩ B’**

**(ii) (A∩B)’ = A’ U B’**

Solution:

U = {1, 2, 3, 4, 5, 6, 7, 8, 9} A = {2, 4, 6, 8}, B = {2, 3, 5, 7}

(i) (AUB) = {2, 3, 4, 5, 6, 7, 8}

(AUB)’ = {1, 9}

A’ = {1, 3, 5, 7, 9} ; B’ = {1, 4, 6, 8, 9}

A’ ∩ B’ = { 1, 9}

Thus, (AUB)’ = A’ ∩ B’

(ii) A∩B = {2}

(A∩B)’ = {1, 3, 4, 5, 6, 7, 8, 9}

A’ = {1, 3, 5, 7, 9} ; B’ = {1, 4, 6, 8, 9}

A’ U B’ = {1, 3, 4, 5, 6, 7, 8, 9}

(A∩B)’ = A’ U B’

**Draw appropriate Venn diagram for each of the following:**

**(i) (AUB)’**

**(II) A’∩ B’**

**(iii) (A∩B)’**

**(iv) A’UB**’

Solution:

(i) (AUB)’

(II) A’∩ B’

(iii) (A∩B)’

(iv) A’UB’

**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 ?**

Solution:

A’ is the set of all equilateral triangles.

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

**(i) AUA’**

**(ii) Φ’∩ A =**

**(iii) A∩A’=**

**(iv) U’∩A =**

Solution:

(i) AUA’ = U

(ii) Φ’∩ A = U∩A = A

Therefore, Φ’∩ A = A

(iii) A∩A’= Φ

(iv) U’∩A = Φ∩ A = Φ

Therefore U’∩ A = Φ