All about Relation, Reflexive Relation, Symmetric Relation, Transitive relation and Equivalence Relation

What is a relation?

A relation R defined in a set A is a subset of A ×A  

Empty Relation: A relation R in a set A is called empty relation, if no element of A is related to any element of A, i.e., R = = A × A.

Universal Relation:  A relation R in a set A is called universal relation, if each element of A is related to every element of A, i.e., R = A × A.

Both the empty relation and the universal relation are also called trivial relations.

Reflexive relation: A relation in a set A is said to be reflexive if and only if           (a, a) ∈ R for every a ∈ set A.

Symmetric Relation:  A relation R on a set A is said to be symmetric if and only if (a, b) ∈ R implies that (b, a) is also belonging to R where a, b ∈ A.

Transitive relation: A relation R on a set A is said to be transitive if and only if       (a, b) ∈ R and (b, c) ∈ R implies that (a, c) is also belonging to R where a, b, c ∈ A.

Equivalence relation: A relation R on a set A is said to be equivalence relation if and only if it is reflexive, symmetric and transitive.