Question

If A = {x : x = 2${}^n$, 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.
$\left[5\right]$

Answer 1

A = {x : x = 2${}^n$, n ∈ W, n < 4}
⇒ x = 2${}^0$
= 1
x = 2${}^1$
= 2
x = 2${}^2$ = 4
x = 2${}^3$ = 8
∴ A = {1, 2, 4, 8}$\left[2\right]$
B = {x : x = 2n, n ∈ N and n ≤ 4}
x = 2 $\times$ 1 = 2
x = 2 $\times$ 2 = 4
x = 2 $\times$ 3 = 6
x = 2 $\times$ 4 = 8
∴ B = {2, 4, 6, 8}
C = {0, 1, 2, 5, 6}$\left[2\right]$
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$\left[1\right]$