Why $(a,b)\mid a+b\le3$ is not reflexive and is symmetric?

Question

Why $(a,b)\mid a+b\le3$ is not reflexive and is symmetric?

I read because a€ Z so by counter example $(5,5)$, $5+5$ is not less than or equal to $3$ So it's not reflexive

But why it's symmetric ? I think because for all a,b €Z So $3+4$ is not less than or equal to $3$ So it is not symmitric

Answer 1

Symmetry does't require that every pair of integers be related.

Symmetry requires ONLY that IF $(a, b)\in R$, THEN $(b, a)\in R$.

We have an implication defining symmetry: $\quad p: (a, b)\in R;\quad q: (b, a) \in R$ where $$p \rightarrow q$$

If $(a, b) \notin R$, that's fine. If our premise $p$ is false, the implication is automatically true. So your example of $3$ and $4$ doesn't present a problem, since $(3, 4)\notin R$. It is only for those pairs $(x, y)\in R$ that are related that we have to ensure that $(y, x)$ is also in the relation.

Answer 2

It is symmetric if for all $a,b$, $$a + b \leq 3 \Leftrightarrow b + a \leq 3$$

Proving whether or not is symmetric in this case boils down to determining whether $b + a \leq 3$ if you know $a + b \leq 3$ (and vice versa).

Answer 3

Symmetric means if $aRb$, then $bRa$, which is the case here.