Is this relation transitive? $S = \{1,2,3,4\}, R = \{(x,y) | x - y \text{ is even and } x - y \geq 0 \}$

Question

This is my first attempt to make up a relation that is transitive, reflexive, but not symmetric. I can't find a counterexample. There are only a few examples, one being:

$$(3,1) \text{ and } (1,1) \Rightarrow (3,1)$$

Which is obviously true...

Answer 1

$(x,y) \wedge (y,z)\to x-z=(x-y)+(y-z)$ is even and nonnegative, so, yes, $R$ is transitive.

Answer 2

What is your question precisely? You can't find a counterexample for what?

If you're seeking for an example of a relation that is transitive and reflexive, yet not symmetric, then every non-strict partial order is an example. The relation mentioned in the title is of course not symmetric and transitive as well

Answer 3

Yes, because a sum of two even numbers is even, and the sum of two nonnegative is nonnegative. So if $x-y$ and $y-z$ are even and nonnegative so is $x-z = (x-y)+(y-z)$.