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

What are your thoughts on this problem? Can you test some of the properties to see if they are satisfied for this relation? — Dan Rust, Feb 27 '15 at 13:58

score 1 · Answer 1 · answered Feb 27 '15 at 14:02

1

Hint:

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

answered Feb 27 '15 at 14:02

Dan Rust

score 0 · Answer 2 · answered Feb 27 '15 at 14:03

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 $mRn$?

2 Answers2