Question

If A = {x : x = 2${}^n$, n $ϵ$ W and n < 4}, B = {x: x = 2 n, n $ϵ$ N and n $\le$ 4} and C = {0, 1, 2, 5, 6}, then verify the associative property of intersection of sets. [5 marks]

Answer 1

A = {x : x = $x=2^n$, n $ϵ$ W, n < 4}

$\Rightarrow x=2^0=2$

$x=2^1=2$

$x=2^2=4$

$x=2^3=8$

$\therefore A=\{1,2,4,8\}$ [1 mark]

B = {x : x = 2n, n $ϵ$ N and n $\le$ 4}

$\Rightarrow x=2\times 1=2$

$x=2\times 2=4$

$x=2\times 3=6$

$x=2\times 4=8$

$\therefore B=\{2,4,6,8\}$ [1 mark]

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

Associative property of intersection of sets

A $\cap$ (B $\cap$ C) = (A $\cap$ B) $\cap$ C

B $\cap$ C = {2, 6} [1 mark]

A $\cap$ (B $\cap$ C) = {1, 2, 4, 8} $\cap$ {2, 6} = {2} ...(1) [½ mark]

A $\cap$ B = {1, 2, 4, 8} $\cap$ {2, 4, 6, 8} = {2, 4, 8} [1 mark]

(A $\cap$ B) $\cap$ C = {2, 4, 8} $\cap$ {0, 1, 2, 5, 6} = {2}... (2) [½ mark]

From (1) and (2), It is verified that A $\cap$ (B $\cap$ C) = (A $\cap$ B) $\cap$ C