I am studying a little bit of set theory, and one of the questions in the book (in Efe A. Ok's real analysis book) asked to show that $\dim(X,\succeq)\leq |X^2|$, where $X$ is a finite set and $(X,\succeq)$ is a preorder.
Here is the full context:
I'm having a little trouble interpreting the notation $|X^2|$. Would it mean the "total number of possible combinations of elements of $X^2$" or would it be 2, as in $|\mathbb{R}^2|$.
Thanks for helping!
For any set $S$, we have the following two notations: $$|S|=\text{the cardinality of }S\qquad\qquad S^2=S\times S=\{(a,b):a,b\in S\}$$ Thus, $|X^2|$ is the cardinality of the set of ordered pairs of elements of $X$. which, incidentally, is equal to $|X|^2$. For example, if $X$ were a set with $2$ elements, then $|X^2|=4$.