Let $A$ and $B$ be sets, then if $A \subseteq B$ then $A \cap B = A$.
How do I prove the above? I'm new to proving, please help!
What are the steps I need to create a proof? I'm really lost.
Now I know that the above is true (I can visualize it, and I can even draw a Venn diagram for it) but I don't really follow on formal proofs.
How should I know what to do? Do I use a let $x$ for $A$? or something else?
Best Answer
To prove two sets are equivalent we need to prove that they contain eachother. By definition of the intersection we have $A\cap B\subseteq A$. Now for the opposite inclusion. Choose $x\in A$. Then, since $A\subseteq B$ we have $x\in B$ and thus $x\in A\cap B$, which implies that $A\subseteq A\cap B$. We conclude that $A = A\cap B$.
To get an intuitive feeling for sets I always draw circles on paper and look at the intersections and unions;)