# 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:

,

,

,

.

**EXAMPLE2:** Let and . Since , then . Let’s show all the elements of this relation on the cartesian coordinate plane:

**DEFINITION2:** Let , be two sets and . The relation defined as

is called the inverse or converse relation of . If , then the inverse of is defined by .

**EXAMPLE3:** Find the inverses of the relations in Example1:

,

,

,

.

**DEFINITION3:** Let be three sets and , be two relations. The relation defined as

is called the composition of the relations and .

**EXAMPLE4:** , , , , , and . Hence:

,

,

,

.

**DEFINITION4:** Let be a set and . (Note that the domain and the codomain of the relation are equal. The following definitions can be given for the relations over a set . It can’t be given for the relations between two difference sets)

**a)** If for all , then the relation is called “reflexive”.

**b)** If it holds “” for , then the relation is called “symmetric”.

**c)** If it holds “” for , then the relation is called “antisymmetric”.

**d)** If it holds “” for , then the relation is called “transitive”.

**DEFINITION5:** Let be a set and . If the relation is reflexive, antisymmetric and transitive, then the relation is called a "partial order relation" and denoted by in general. If "" is a partial order relation over a set , then is called "partially ordered set" or shortly "poset".

**DEFINITION6:** Let be a set and . If the relation is reflexive, symmetric and transitive, then the relation is called an "equivalence relation" and denoted by in general.