Question

Given $A=$ {$x\in N$ : $x<6$}, $B=$ {$3,6,9$} and $C=$ {$x\in N$ : $2x-5\le 8$}. Show that :

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

Answer 1

$A=\{1,2,3,4,5\}$

$B=\{3,6,9\}$

$C=\{x$ $\in$ $N:2x-5$ $\le 8$$\}$

To find $C$,

$2x-5=8$

$2x=8+5$

$x=$$\frac{13}{2}$ - not a natural numbers

$C=\{1,2,3,4,5,6\}$

To prove

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

$B$ $\cap$ $C=\{3,6\}$

$A$ $\cup$ $(B$ $\cap$ $C)=\{1,2,3,4,5,6\}$ $\to$ (1)

$A$ $\cup$$B=\{1,2,3,4,5,6,9\}$

$A$ $\cup$ $C=\{1,2,3,4,5,6\}$

$(A$ $\cup$ $B)$ $\cap$$(A$ $\cup$$C)=\{1,2,3,4,5,6\}$ $\to$ (2)

(1) and (2) are equal

Hence prove that $A\cup$ $(B$ $\cap$ $C)=(A$ $\cup$ $B)$ $\cap$$(A$ $\cup$$C)$.