Definition 3.5.4 (Cartesian product). If \(X\) and \(Y\) are sets, then we define Cartesian product \(X\times Y\) to the collection of ordered pairs, whose first component lies in \(X\) and second component lies in \(Y\), thus $$ X\times Y=\{(x,y):x\in X,y\in Y\} $$ or equivalently $$ a\in(X\times Y)\iff(a=(x,y) \text{ for some } x\in X\text{ and }y\in Y). $$