Q:

Understanding the Empty set and set theory via proof

This question has already been asked. see above link. 

A:

Recall the definitions.

$X\cap Y$ is the set of all $a$ such that $a\in X$ and $a\in Y$.
$X - Y$ is the set of all $a$ such that $a\in X$ and $a\notin Y$. (Also denoted by $X\setminus Y$ sometimes.)
$X=Y$ if for every $a$ we have that if $a\in X$, then $a\in Y$ and if $a\in Y$ then $a\in X$. This translates to the statement that $X\subseteq Y$ and $Y\subseteq X$.

So we have to show equality between two sets. One of them is empty, and the empty set is always a subset of every set, so it suffices to show that there are no elements to the set in the left hand side of the equation.
So you have to prove that given any element $a$, it is impossible that $a$ is in $Y$ and in $X-Y$. How should you do that? Unfold the definition. If it is in both sets, then it satisfies both requirements. Can you see why this is impossible?

