# Relation

**DEFINITION1:** Let and be two sets. Any subset of the cartesian product is called a relation with domain and codomain . While some sources are giving the definition of relation, they assume and it’s said the emptyset being a subset of isn’t a relation. However, the assumption “the emptyset is a relation” is not a problem for any branch of the mathematics. On the contrary, the assumption “the emptyset is a relation” plays an important role in some branch of the mathematics.

If and are two sets with and elements respectively, then the cartesian product has elements. Since a relation with domain and codomain is an element of the power set and the number of the elements of the power set of a set with elements is , then the number of all the relations with domain and codomain is . If and at least one of and is infinite set, then the number of all the relations with domain and codomain is also infinity.

Let be a relation not being the emptyset. The statement is read “x is R-related to y” and is denoted by or .

**EXAMPLE1:** Let and . Since has 3 elements and has 2 elements, the number of all the relations with domain and codomain is . We can give some of these relations:

,