Suppose that  $A=\{1,2,3\}$ and the relation $f=\{(1,1),(2,2),(3,3)\}$ defined on the set  A then the relation  $f$ is

Let $A$ be any non empty set. 

1. A relation $R$ is said to reflexive relation if $\left(a,a\right)\in R$ for all $a\in A$, 
2. A relation $R$is said to Symmetric relation if $\left(a,b\right)\in R\Rightarrow \left(b,a\right)\in R$for all $a,b\in A$
3. A relation $R$is said to be Transitive relation if either $\left(a,b\right),\left(b,c\right)\in R\Rightarrow \left(a,c\right)\in R$or $\left(a,b\right)\in R, and b has no image in A$
4. A relation $R$is said to be equivalence relation if it is Reflexive, Symmetric, Transitive. 

Given $A=\left\{1,2,3\right\}$ and the relation is defined as $f=\{(1,1),(2,2),(3,3)\}$

This is Reflexive, symmetric, transitive relations, hence it is equivalence relation.