7. Five mark exercise problems II

If \(A = \{x: x = 2^n, n \in W \ \text{and} \ n < 4\}\), \(B = \{x: x = 2n, n \in N \ \text{and} \ n \leq 4\}\) and \(C = \{0, 1, 2, 5, 6\}\), then verify the associative property of intersection of sets.

 

Proof:

 

\(A =\) $\left\{i,i,i,i\right\}$

 

\(B =\) $\left\{i,i,i,i\right\}$

 

\(C = \{0, 1, 2, 5, 6\}\)

 

To prove: \(A \cap (B \cap C) = (A \cap B) \cap C\)

 

\(B \cap C =\) $\left\{i,i\right\}$

 

\(A \cap (B \cap C) =\) $\left\{i\right\}$ ---- (\(1\))

 

\(A \cap B =\) $\left\{i,i,i\right\}$

 

\((A \cap B) \cap C =\) $\left\{i\right\}$ ---- (\(2\))

 

From equations (\(1\)) and (\(2\)), we have \(A \cap (B \cap C) = (A \cap B) \cap C\).

 

Hence, we proved.