7. Five mark example problems III

If $P = \{x: x \in W \ \text{and} \ 0 < x < 10\}$, $Q = \{x: x = 2n + 1, n \in W \ \text{and} \ n < 5\}$ and $R = \{2, 3, 5, 7, 11, 13\}$, then verify $P - (Q \cap R) = (P - Q) \cup (P - R)$.

 

Proof:

 

$P =$

 

$Q =$

 

$R = \{2, 3, 5, 7, 11, 13\}$

 

$Q \cap R =$

 

$P - (Q \cap R) =$  ---- ($1$)

 

$P - Q =$

 

$P - R =$

 

$(P - Q) \cup (P - R) =$  ---- ($2$)

 

From equations ($1$) and ($2$), $P - (Q \cap R) = (P - Q) \cup (P - R)$.

 

Hence, we proved.