Lets say I have the set $X_n=\{1,2,3,...,n\}$. Then I have: $X_n \times X_n= (i,j)|i,j \in X_n$
What does it mean when I say let $U$ be the subset of $X_n \times X_n$ where $i<j$? I don't understand how two sets with the same carnality can be compared like this?
On
$X_n \times X_n$ is a collection of pairs of elements of $X_n$.
$U$ is the subset of $X_n \times X_n$ where $i < j$. This means that the first element of the pair must be strictly smaller than the second element of the pair. For example, let us say $n=3$. Then,
$X_n \times X_n = \{(1,1), (1,2), (1,3), (2,1), (2,2), (2,3), (3,1),(3,2),(3,3)\}$: those are all possible pairs of elements of $X_n$.
$U$ then becomes $U = \{(1,2),(1,3),(2,3)\}$, because those are the pairs in which the first element, $i$, is strictly smaller than $j$.
I don't really know what you mean by "comparing". You just defined a set and a subset of that set.
It may help to look at a concrete example. If $n = 5$, then we have: $$ U = \left\{\begin{array}{r} (1, 2), (1, 3), (1, 4), (1, 5), \\ (2, 3), (2, 4), (2, 5), \\ (3, 4), (3, 5), \\ (4, 5) \ \end{array}\right\} $$