Question

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

Answer 1

A=$\{x\in N:x<6\}$ $\{1,2,3,4,5\}$ B=$\{3,6,9\}$ C$\{x\in N:2x-5\le 8\}$ 2x-5 $\le$ $\Rightarrow 2x\le 8+5\Rightarrow 2x\le 13\Rightarrow x\le \frac{13}{2}\Rightarrow x\le 6.5\therefore C=\{1,2,3,4,5,6\}$ (i) $B\cap C=\{3,6,9\}\cap \{1,2,3,4,5,6\}=\{3,6\}\therefore A\cup (B\cap C)=\{1,2,3,4,5\}\cup \{3,6\}=\{1,2,3,4,5,6\}A\cup B=\{1,2,3,4,5\}\cup \{3,6,9\}=\{1,2,3,4,5,6,9\}A\cup C=\{1,2,3,4,5\}\cup \{1,2,3,4,5,6\}=\{1,2,3,4,5,6\}$ $\therefore (A\cup B)\cap (A\cup C)=\{1,2,3,4,5,6,9\}\cap \{1,2,3,4,5,6\}=\{1,2,3,4,5,6\}\therefore A\cup (B\cap C)=(A\cup B)\cap (A\cup C)$ (ii)$B\cup C=\{3,6,9\}\cup \{1,2,3,4,5,6\}=\{1,2,3,4,5,6,9\}A\cap (B\cup C)=\{1,2,3,4,5\}\cap \{1,2,3,4,5,6,9\}=\{1,2,3,4,5\}NowA\cap C=\{1,2,3,4,5\}\cap \{1,2,3,4,5,6\}=\{1,2,3,4,5\}A\cap B=\{1,2,3,4,5\}\cap \{3,6,9\}=\left\{3\right\}\therefore (A\cap B)\cup (A\cap C)=\left\{3\right\}\cup \{1,2,3,4,5\}=\{1,2,3,4,5\}\therefore A\cap (B\cup C)=(A\cap B)\cup (A\cap C)$