Which of the following is not an equivalence relation in Z?

Let R = {$\left(a,b\right)$ : a + b is even integer. a, b$\in$ Z }.

For, a$\in$ Z, a + a = 2a is an even integer.

(a, a) $\in$R,$\forall$ a$\in$ Z

R is reflective.

Let     (a, b) $\in$R $\Rightarrow$ a + b is even integer.

$\Rightarrow$     b + a is even integer   (b, a)$\in$R

$\Rightarrow$R is symmetric

Let                        (a, b),(b, c)$\in$R

$\Rightarrow$  a + b and b + c are even integer

$\Rightarrow$ (a + b) + (b + c) = a + c + 2b is even integer

$\Rightarrow$ (a + c + 2b) - 2b = a + c is even integer.

$\Rightarrow$            (a, c)$\in$ R$\Rightarrow$    R is transitive

R is an equivalence relation.

Let R = {(a, b) : a - b is even integer, a, b $\in$Z}

For a $\in$Z, a -a=0 is an even integer.

                         (a $-$a ) $\in$ R $\forall$a $\in$ Z

R is reflexive.

Let (a, b) $\in$R.   $\Rightarrow$ a -b is even integer.

   $\Rightarrow$ $-$ (a - b) is even integer.

  $\Rightarrow$b - a is even integer. $\Rightarrow$ (b ,a) $\in$R

$\Rightarrow$R symmetric

Let (a, b), (b, c)$\in$R $\Rightarrow$ a- b , b- c are even integers.

                              ( a -b ) + ( b - c)  is even integer.

$\Rightarrow$       a - c is even integer.

$\Rightarrow$           (a , c ) $\in$R

$\Rightarrow$  R is transitive.

 R is an equivalence relation.

Let                        R = {(a, b) : a < b; a, b$\in$ Z }

Let                        a$\in$ Z, $\Rightarrow$ a < a is false.

R is not reflexive.

R is not an equivalence relation.

Let                  R = {( a, b) : a = b, a, b$\in Z \}$ clearly it is an equivalence relation on Z