What is the value of the intersection of X and the set containing X? How to calculate X $\cap$ $\{X\}$ for finite sets to develop an intuition for intersections?
If $X$ = $\{$1,2,3$\}$, then what is $X$ $\cap$ $\{X\}$? 

As far as developing intuition for intersection, the idea of $A \cap B$ are the elements that $A$ and $B$ both have in common. So if we're looking at $X \cap \left\{ X \right\}$ where $X = \left\{ 1,2,3 \right\}$ then it is a matter of 
$$X \cap \left\{ X \right\} = \left\{ 1,2,3 \right\} \cap \left\{ \left\{ 1,2,3 \right\} \right\}.$$
However, there is a rather subtle difference here between the left and right side of the intersection. The left side, $\left\{ 1,2,3\right\} = X$, is the set at hand. Where the right side, $\left\{ X \right\}$ is viewing the set $X$ as an element, which is different than $X$ itself, so they have nothing in common. Hence,
$$X \cap \left\{ X \right\} = \emptyset.$$
