Equivalence relation on a set of integers

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 1

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.

Answer 2

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 $mRn$?