Cartesian product

The Cartesian product of two sets \[A\] and \[B\], denoted \[A\times B\] is the set of all ordered pairs \[(a,b)\], that is \[A\times B=\left\{ (a,b)\mid a\in A \,\,\text{and}\,\,b\in B \right\}\]. For instance, assuming \[A=\left\{ 1,2,3 \right\}\] and \[C=\left\{ 4,5 \right\}\], then \[A\times C=\left\{ (1,4),(1,5),(2,4),(2,5),(3,4),(3,5) \right\}\].

Note that ordered pairs are characterized by the following property: \[(a,b)=(c,d)\] if and only if \[a=c\] and \[b=d\].