Question

If $A$ and $B$ are subsets of the universal set $U$, then show that

i $A\subset A\cup B$

ii $A\subset B\iff A\cup B=B$

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

Answer 1

Given : $A$ and $B$ are subsets of the universal set $U$

To prove : $A\subset A\cup B$

We know, $A\cup B=\{x|x\in Aorx\in B\}$

Thus, $x\in A\Rightarrow x\in A\cup B$

Hence, $A\subset A\cup B$


To prove : $A\subset B\iff A\cup B=B$

If $A\subset B$ is given

Let $x\in A\cup B$

$\Rightarrow x\in Aorx\in B$

$\Rightarrow x\in B$ $[\because A\subset B]$

$\Rightarrow A\cup B\subset B$ ...$\left(1\right)$

But we know $B\subset A\cup B$ ...$\left(2\right)$

From $\left(1\right)$ and $\left(2\right)$, we get $A\cup B=B$

Now if $A\cup B=B$

Let $y\in A$

$\Rightarrow y\in A\cup B$

$\Rightarrow y\in B$ $[\because A\cup B=B]$

$\Rightarrow A\subset B$

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


To prove :

$(A\cap B)\subset A$

Let $A\cap B=\{x|x\in Aandx\in B\}$

$\Rightarrow x\in A\cap B$

$\Rightarrow x\in A$

Hence, $A\cap B\subset A$