Question

If A = {x : x = 2ⁿ, n ∈ W and n < 4}, B = {x : x = 2n, n ∈ N and n ≤ 4} and C = {0, 1, 2, 5, 6}, then verify the associative property of intersection of sets

Answer 1

A = {x : x = 2ⁿ, n ∈ W, n < 4}

⇒ x = 2° = 1

x = 2¹ = 2

x = 2² = 4

x = 2³ = 8

∴ A = {1, 2, 4, 8}

B = {x : x = 2n, n ∈ N and n ≤ 4}

⇒ x = 2 × 1 = 2

x = 2 × 2 = 4

x = 2 × 3 = 6

x = 2 × 4 = 8

∴ B = {2, 4, 6, 8}

C = {0, 1, 2, 5, 6}

Associative property of intersection of sets

A ∩ (B ∩ C) = (A ∩ B) ∩ C

B ∩ C = {2, 6}

A ∩ (B ∩ C) = {1, 2, 4, 8} ∩ {2, 6}

= {2}   ...(1)

A ∩ B = {1, 2, 4, 8} ∩ {2, 4, 6, 8}

= {2, 4, 8}

(A ∩ B) ∩ C = {2, 4, 8} ∩ {0, 1, 2, 5, 6}

= {2}   ...(2)

From (1) and (2)

It is verified that A ∩ (B ∩ C) = (A ∩ B) ∩ C