Question

For any two sets A and B, prove that

(i) $B\subset A\cup B$

(ii) $A\cap B\subset A$

(iii) $A\subset B\Rightarrow A\cap B=A$

Answer 1

(i) $B\subset A\cup B$

Let $xϵA$

$\Rightarrow xϵAorxϵB\Rightarrow xϵA\cup B$

Hence , $B\subset A\cup B$

(ii) $A\cap B\subset A$

Let $xϵA\cap B$

$\Rightarrow xϵAandxϵB\Rightarrow xϵA$

Hence, $A\cap B\subset A$

(iii) $A\subset B\Rightarrow A\cap B=A$

If $A\subset B$

Let $xϵA\cup B$

$\Rightarrow xϵAorxϵB\to xϵB$ [$\therefore A\subset B$]

$\Rightarrow A\cup B\subset B$ ....(i)

But $B\subset A\cup B$ ....(ii)

From Eqs. (i) and (ii)

$A\cup B=B$

If $A\cup B=B$

Let $yϵA$

$\Rightarrow yϵA\cup B\Rightarrow yϵB$ [$\therefore A\cup B=B$]

$\Rightarrow A\subset B$

Hence, $A\subset B\iff A\cup B=B$