Question

If $A=\{-3,-1,0,4,6,8,10\},B=\{-1,-2,3,4,5,6\},andC=\{-6,-4,-2,2,4,6\}$ show that $A\cup (B\cap C)=(A\cup B)\cap (A\cup C)$

Answer 1

The given sets are $A=\{-3,-1,0,4,6,8,10\}$, $B=\{-1,-2,3,4,5,6\}$ and $C=\{-6,-4,-2,2,4,6\}$

Consider the following:

$A\cup (B\cap C)=\{-3,-1,0,4,6,8,10\}\cup \left[\right\{-1,-2,3,4,5,6\}\cap \{-6,-4,-2,2,4,6\left\}\right]$

$=\{-3,-1,0,4,6,8,10\}\cup \{-2,4,6\}=\{-3,-2,-1,0,4,6,8,10\}......\left(1\right)$

$(A\cup B)\cap (A\cup C)=\left[\right\{-3,-1,0,4,6,8,10\}\cup \{-1,-2,3,4,5,6\left\}\right]\cap \left[\right\{-3,-1,0,4,6,8,10\}\cup \{-6,-4,-2,2,4,6\left\}\right]$

$=\{-3,-2,-1,0,3,4,5,6,8,10\}\cap \{-6,-4,-3,-2,-1,0,2,4,6,8,10\}=\{-3,-2,-1,0,4,6,8,10\}......\left(2\right)$

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

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