Question

Let $A,B$ and $C$ be sets. Then show that $A\cap (B\cup C)=(A\cap B)\cup (A\cap C)$

Answer 1

To prove: $A\cap (B\cup C)=(A\cap B)\cup (A\cap C)$

Solving $L.H.S$

Let $x\in A\cap (B\cup C)$

$\Rightarrow x\in A$ and $x\in (B\cup C)$

$\Rightarrow x\in A$ and $(x\in Borx\in C)$

$\Rightarrow (x\in Aandx\in B)$ or $(x\in Aandx\in C)$

$\Rightarrow x\in (A\cap B)$ or $x\in (A\cap C)$

$\Rightarrow x\in (A\cap B)\cup (A\cap C)$

$\Rightarrow A\cap (B\cup C)\subset (A\cap B)\cup (A\cap C)$ $\dots \left(1\right)$

Let $y\in (A\cap B)\cup (A\cap C)$

$\Rightarrow y\in (A\cap B)$ or $y\in (A\cap C)$

$\Rightarrow (y\in Aandy\in B)$ or $(y\in Aandy\in C)$

$\Rightarrow y\in A$ and $(y\in Bory\in C)$

$\Rightarrow y\in A$ and $y\in (B\cup C)$

$\Rightarrow y\in A\cap (B\cup C)$

$\Rightarrow (A\cap B)\cup (A\cap C)\subset A\cap (B\cup C)$ $\dots \left(2\right)$

We know if $A\subset B$ and $B\subset A$

Then, $A=B$

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

Hence proved.