Cartesian Product of Two Sets

Let A and B be two non-empty sets. The CARTESIAN PRODUCT of A and B, denoted $A\times B$, is defined as the set of ALL ordered pairs $(a,b)$ such that $a\in A$ and $b\in B$: $A\times B = \{(a,b)/a\in A, b\in B\}$. For example, if $A=\{1,2\}$ and $B=\{a,b,c\}$, then $A\times B = \{(1,a),(1,b),(1,c),(2,a),(2,b),(2,c)\}$ — every element of A paired with every element of B, in that order. If either $A=\phi$ or $B=\phi$, then by definition $A\times B=\phi$ (there are no elements to pair up). …