Disproving equality of cartesian products We are to disprove the statement $X \times Y = Y \times X \iff X = Y$ but I can't think of an example where this would be false. If $X = Y$, then wouldn't the Cartesian product be the same in either direction?

The important point is that a Cartesian product is a set of ordered pairs.  So if $X$ (say) contains an element $a$ which is not in $Y$, then in $X \times Y$ the element $a$ will appear in ordered pairs only as the first item of the pair, while in $Y \times X$ it will appear only as the second item of the pair.  So no pair containing $a$ in $X \times\ Y$ can ever equal any pair in $Y \times X$, even one also containing $a$.
