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.