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

$A \subset B \Leftrightarrow A \cup B = B$

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Solution

If $A \subset B$

Let $x ϵ A \cup B$

$\Rightarrow x ϵ A o r x ϵ B \Rightarrow x ϵ B$

$\Rightarrow A \cup B \subset B$

But $B \subset A \cup B$

From Eqs. (i) and (ii),

$A \cup B$ = B

If $A \cup B$ = B

Let y $ϵ A$

$\Rightarrow$ y $ϵ A \cup B$ $\Rightarrow$ y $ϵ B$ $[∵ A \cup B = B]$

$\Rightarrow A \subset B$

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

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