Which of the following are not equivalence relations on $I$?

Option A, $aRb$ if $a+b$ is an even integer.

Let $a,b,c\in I$
Clearly, $a+a=2a$ is an even integer.
Hence, $aRa$ for all $a\in I$

Now, let $aRb$ 
$\Rightarrow a+b$ is an even integer.
or $b+a$ is an even integer.
$\Rightarrow bRa$

Next, let $aRb,bRc$
$\Rightarrow a+b$ in an even integer and $b+c$ is an even integer
$\Rightarrow a$ and $b$ both are even or both are odd . Also, $b$ and $c$ both are even or both are odd.
Hence, if $a$ and $b$ are even .So $c$ is also even . Hence, $a+c$ is even.
And if $a$ and $b$ are odd.So, $c$ is also odd . Again $a+c$ is even.
Hence, $aRc$

Hence, option A is equivalence relation.

Option B, $aRb$ if $a-b$ is an even integer.

Let $a,b,c\in I$
Clearly, $a-a=0$ is an even integer.
Hence, $aRa$ for all $a\in I$

Now, let $aRb$ 
$\Rightarrow a-b$ is an even integer.
or $-(a-b)$ is also an even integer.
$\Rightarrow bRa$

Next, let $aRb,bRc$
$\Rightarrow a-b$ in an even integer and $b-c$ is an even integer
$a+c=a-b+b-c=$ even+even 
Hence, $a+c$ is even.
Hence, $aRc$

Hence, option $B$ is equivalence relation.

Option C $aRb$ if $a<b$

Let $a,b,c\in I$
Clearly,symmetry does not hold here 
Now, let $aRb$ 
$\Rightarrow a<b$ 
But it does not implies $b<a$

Hence, option C is not an equivalence relation.

Option D, $aRb$ if $a=b$ 

Let $a,b,c\in I$
Clearly, $a=a$ 
Hence, $aRa$ for all $a\in I$

Now, let $aRb$ 
$\Rightarrow a=b$ 
or $b=a$ is an even integer.
$\Rightarrow bRa$

Next, let $aRb,bRc$
$\Rightarrow a=b$and $b=c$ 
$\Rightarrow a=b=c$
Hence, $aRc$

Hence, option D is equivalence relation.