Suppose $A$ and $B$ are sets. Prove that $A \subseteq B$ if and only if $A \cap B = A$.
Here's how I see it being proved.
If $A$ and $B$ are sets,and the intersection of $A$ and $B$ is equal to $A$, then the elements in $A$ are in both the set $A$ and $B$. Therefore, the set of $A$ is a subset of $B$ since all the elements are contained in the interesection of sets $A$ and $B$ are equal to $A$.
Can I prove it that way?
Best Answer
Your proof is almost perfect and let me rectify it a bit:
Let $A$ and $B$ be two sets. The intersection of $A$ and $B$ is equal to $A$, is equivalent to the elements in $A$ are in both the set $A$ and $B$ which's also equivalent to the set of $A$ is a subset of $B$ since all the elements of $A$ are contained in the intersection of sets $A$ and $B$ are equal to $A$.