Question

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

• A. $aRb$ if $a+b$ is an even integer
• B. $aRb$ if $a-b$ is an even integer
• C. $aRb$ if $a<b$
• D. $aRb$ if $a=b$

Answer 1

The correct option is C $aRb$ if $a<b$

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.