Question

If A = {x : x = 6n, n ∈ W and n < 6}, B = {x : x = 2n, n ∈ N and 2 < n ≤ 9} and C = {x : x = 3n, n ∈ N and 4 ≤ n < 10}, then show that A – (B ∩ C) = (A – B) ∪ (A – C)

Answer 1

A = {x : x = 6n, n ∈ W, n < 6}

x = 6n

n = {0, 1, 2, 3, 4, 5}

⇒ x = 6 × 0 = 0

x = 6 × 1 = 6

x = 6 × 2 = 12

x = 6 × 3 = 18

x = 6 × 4 = 24

x = 6 × 5 = 30

∴ A = {0, 6, 12, 18, 24, 30}

B = {x : x = 2n, n ∈ N, 2 < n ≤ 9}

n = {3, 4, 5, 6, 7, 8, 9}

x = 2n

⇒ x = 2 × 3 = 6

2 × 4 = 8

2 × 5 = 10

2 × 6 = 12

2 × 7 = 14

2 × 8 = 16

2 × 9 = 18

∴ B = {6, 8, 10, 12, 14, 16, 18}

C = {x : x = 3n, n ∈ N, 4 ≤ n < 10}

N = { 4, 5, 6, 7, 8, 9}

x = 3 × 4 = 12

⇒ x = 3 × 5 = 15

x = 3 × 6 = 18

x = 3 × 7 = 21

x = 3 × 8 = 24

x = 3 × 9 = 27

x = 2 × 9 = 18

∴ C = {12, 15, 18, 21, 24, 27}

A – (B ∩ C) = (A – B) ∪ (A – C)

L.H.S = R.H.S

B ∩ C = {12, 18}

A – (B ∩ C) = {0, 6, 12, 18, 24, 30} – {12, 18}

= {0, 6, 24, 30}   ...(1)

(A – B) = {0, 24, 30}

(A – C) = {0, 6, 30}

(A – B) ∪ (A – C) = {0, 6, 24, 30}  ...(2)

From (1) and (2), it is verified that

A – (B ∩ C) = (A – B) ∪ (A – C).