Concept of a relation

Let \(A\), \(B\) be two sets, and we define the product \(A \times B\) is the set of all pairs \((a,b)\), where \(a \in A, b \in B\).

For example, if \(A = \{1,2\}\) and \(B = \{4,5,6\}\) then

\[A \times B = \{(1,4), (1,5), (1,6), (2,4), (2,5), (2,6)\}.\]

With this concept of product, we can now define what is a relation \(R\) between the set \(A\) and the set \(B\).

A relation \(R\) between sets \(A\) and \(B\) is just a subset of \(A \times B\).