If A = {x : x = 2n, n ∈ W and n < 4}, B = {x : x = 2 n, n ∈ N and

1.	If A = {x : x = 2n, n ∈ W and n < 4}, B = {x : x = 2 n, n ∈ N and n ≤ 4} and C = {0, 1, 2, 5, 6}, then verify the associative property of intersection of sets.
Answer» A = {x : x = 2ⁿ, n ∈ W, n < 4} ⇒ x = 2⁰ = 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 x 1 = 2 x = 2 x 2 = 4 x = 2 x 3 = 6 x = 2 x 4 = 8 ∴ B = {2, 4, 6, 8} C = {0, 1, 2, 5, 6} Associative property of intersection of set 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

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

Answer»

A = {x : x = 2ⁿ, n ∈ W, n < 4}

⇒ x = 2⁰ = 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 x 1 = 2

x = 2 x 2 = 4

x = 2 x 3 = 6

x = 2 x 4 = 8

∴ B = {2, 4, 6, 8}

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

Associative property of intersection of set

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