Say we have a relation $R$ on $\mathbb{Z} \times \mathbb{Z}$ such that $(a, b) R (c, d)$ if $a^2 + b^2 \leq c^2 + d^2$
So to prove that $R$ is not an equivalence relation we need to show that $R$
- Is not one of reflexive, symmetric or transitive
And to prove that $R$ is not a partial order we need to show that $R$
- Is not one of reflexive, anti-symmetric or transitive
I'm practicing relation type questions, however, my current experience has mainly been with two variables (one on each side of the relation,) so, I'm struggling a bit with this question.
My attempt so far is as follows
$R$ is reflexive as $(a, a) R (a, a)$ because $a^2 + a^2 \leq a^2 + a^2$
$R$ is not symmetric as $a^2 + b^2 \leq c^2 + d^2$ does not imply that $c^2 + d^2 \leq a^2 + b^2$
So as $R$ is not symmetric it cannot be an equivalence relation.
At this point, I'm a bit stuck. I'm not sure how to test if $R$ is transitive or anti-symmetric.