Consider the relation on the set of integer R={(a, b) /a=b+1} check whether it is equivalence relation

A={1,2 3,4,5,6}

{(1,1)(1,2)(1,3)(1,4)(1,5)(1,6)(2,1)(2,2)(2,3)(2,4)(2,5)(2,6)(3,1)(3,2)(3,3)(3,4),(3,5)(3,6)}

R={(2,1)(3,2)}

1

$\forall a \isin A$ $,(a,a)\notin R$

(1,1)(2,2)(3,3)$\notin$ R

This is not reflexive

2 (a,b)$\isin$ R but (b,a)$\notin$ R

(2,1)$\isin$ R but (1,2)$\notin$ R

SO this is antisymmetric

3 (a,b)$\isin$ R

(b,c)$\isin$ R

(c,a)$\isin$ R

This is a transitive property.

So this is not an equivalance relation.