Question: I was wondering if the relation $X$ would be an equivalence relation only if the result is an even number. 

For example the relation $X$ is given by $a\ X\ b$ only if $a+b$ is even. 

Would this be considered an equivalence relation making is reflexive, symmetric and transitive?

Thanks

Answer: 
Hint: 

* Is the sum of two odd numbers even or odd?
* Is the sum of two even numbers even or odd?
* Is the sum of an even number and an odd number even or odd?

Given the answers to the above, split the testing of each property into different cases.
Use the definitions:

For example:

* $R$ is reflexive if, for every $x$, we have $xRx$. Take an arbitrary number $n$. Can you prove that $nRn$? You can prove this by showing that $n+n$ is even. If you cannot prove this, can you find an example where it is not true?
* $R$ is symmetric if, for any $x,y$ such that $xRy$, we have $yRx$. Take any two $n,m$ such that $nRm$. Can you then prove that $mRn$?