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We know that $A \subseteq B \Rightarrow A \cup B = B$. Is the theorem valid in the other direction $\Leftarrow$?

I thought of it to be intuitive that the $\Leftarrow$ direction should be valid too. Is it?

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up vote 6 down vote accepted

Suppose $A\cup B=B$ and let $a\in A$. Then clearly $a\in A\cup B$ and thus $a\in B$. This shows that $A\subseteq B$.

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This is obvious since $A\subseteq A\cup B = B$.

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enter image description here

Given : $A\cup B = B$

Since $A\cup B$ contains all elements of $A$, $B$ should also contain all elements of $A$. Hence $A\subseteq B$

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