The question:
Let R be a relation over the positive integers defined as follows:
$ \{ (a,b) \mid $ gcd$(a,b) > 1 $ but $ a \nmid b $ and $b \nmid a \} $
Determine whether or not R satisfies the following properties. reflexive, irreflexive, symmetric, anti-symmetric, and transitive. Give a brief justification for each of your answers.
My attempt at figuring out what the relation statement meant:
$(a,b)$ exists such that the greatest common divisor of $(a,b)$ is greater than 1. But, a doesn't divide into b and b doesn't divide into a.
Almost. I wouldn't say "$(a,b)$ exists".
This relation is the set of pairs with gcd greater than $1$ where neither divides the other. So it contains $(6,15)$ but not $(6,18)$ and of course not $(6,7)$. (Writing down a few examples is always a good way to test your understanding.)