2. Two mark example problems II

1. If \(A =\) $\left\{-\frac{1}{2},0,\frac{1}{4},\frac{3}{4},2\right\}$ and \(B =\) $\left\{0,\frac{1}{4},\frac{3}{4},2,\frac{5}{2}\right\}$ and \(C =\) $\left\{-\frac{1}{2},\frac{1}{4},1,2,\frac{5}{2}\right\}$, then verify \(A \cap (B \cap C) = (A \cap B) \cap C\).

 

Proof:

 

\(B \cap C =\)

 

\(A \cap (B \cap C) =\)  ---- (\(1\))

 

\(A \cap B =\)

 

\((A \cap B) \cap C =\)  ---- (\(2\))

 

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

 

Hence, we proved.

 

 

2. If \(A = \{0, 2, 4, 6, 8\}\), \(B = \{x: x \ \text{is a prime number and} \ x < 11\}\) and \(C = \{x : x \in N \ \text{and} \ 5 \leq x < 9\}\) then verify \(A \cup (B \cap C) = (A \cup B) \cap (A \cup C)\).

 

Proof:

 

\(B \cap C =\)

 

\(A \cup (B \cap C) =\)  ---- (\(1\))

 

\(A \cup B =\)

 

\(A \cup C =\)

 

\((A \cup B) \cap (A \cup C) =\)  ---- (\(2\))

 

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

 

Hence, we proved.