How to show a relation is/isn't reflexive, transitive, or symmetric

Question

I was tasked with this:

Define a relation on Z by setting x R y if xy is even.

(a) Give a counterexample to show that R is not reflexive.

How do I go about proving this? Do I express this in terms of predicate logic or something and use laws and rules and the like to prove this? I think I have an idea of what I need to do, though I don't think I have a correct understanding.

I would start by saying assume xy is odd. Then, by definition of odd, xy = 2k + 1, where k is any integer. From here on, I'm stuck. I know reflexivity involves something like x R x but I don't know how to apply that to this problem.

Any help is appreciated, especially before 8am Eastern 2/18/15 when this homework is due.

Answer 1

Saying that $R$ is reflexive means:

for every $x\in\Bbb Z$ we have $x\,R\,x$.

To show that $R$ is not reflexive, you need to give one example where the statement $x\,R\,x$ is false.

In this case "$x\,R\,y$'' means "$xy$ is even", so "$x\,R\,x$" means "$x^2$ is even". Can you find an example of an integer $x$ such that $x^2$ is not even?