Types of Relations

Empty Relation is the relation R in a set A , in which no element of A is related to any element of A, i.e., $$\mathrm{R}=\phi \subset \mathrm{A} \times \mathrm{A}$$.
Universal Relation is the relation R in a set A , in which each element of A is related to every element of A , i.e., $$\mathrm{R}=\mathrm{A} \times \mathrm{A}$$.
Both the empty relation and the universal relation are some times called trivial relations.
Reflexive Relation : $$(a, a) \in R$$, for every $$a \in A$$.
Symmetric Relation : $$\left(a_1, a_2\right) \in R$$ implies that $$\left(a_2, a_1\right) \in R$$, for all $$\mathrm{a}_1, \mathrm{a}_2 \in \mathrm{~A}$$.
Transitive Relation : $$\left(a_1, a_2\right) \in R \quad \&\left(a_2, a_3\right) \in R$$ implies that $$\left(a_1, a_3\right) \in R$$, for all $$a_1, a_2, a_3 \in A$$.
Equivalence Relation : A relation R in a set A is said to be an equivalence relation if $$R$$ is reflexive, symmetric & transitive.
Equivalence class $$\{\mathrm{a}\}$$ containing $$\mathrm{a} \in \mathrm{A}$$ for an equivalence relation R in A is a subset of A containing all elements b related to a .