How do I prove a basic and obvious-looking set relations? I'm a beginner in set theory, but the exercises asking for proof for intuitively obvious set relations like $A\cap A=A$. I don't know where to start. It will be appreciated if there is an example.
p.s. Mathematics consists of proving the most obvious thing in the least obvious way. - Polya
Add. I don't know the reason but I can't use comment. @amWhy.
No proof for $\left(x \in A \;\wedge \; x \in A\right)\ \leftrightarrow \left( x \in A\right)$ ?
 A: $$A\cap A = \{x\mid x \in A \;\text{and}\; x \in A\} = \{x \mid x \in A\} = A$$
That reads as follows: $A\cap A$ is the set of all elements that are in both $A$ and $A$, which is simply the set of all elements in $A$, which is exactly $A$.
A: One of the standard ways to go about this type of equalities consists in proving, in this case, that
$A \cap A \subset A$ and $A \cap A \supset A$ i.e. proving that every element of $A \cap A$ also belongs to $A$ and viceversa. 
A: Perhaps begin by taking an element $a\in A \cap A$. Then since $a$ lies in both $A$ and $A$, then it must lie in $A$. So we have $A \cap A \subseteq A$. The other way is a similar argument. 
A great strategy of showing two sets are equal is to take an arbitrary element of one set and showing that it must lie in the other set. Then do this for the other direction.
A: This might seem like obvious advice, but if you are stuck, always go back to the definition.  I find this helps when trying to prove things that might seem clear.  In general it helps when you dont know where to begin. 
