Is $m\sim n \iff m\leq n$ an equivalence relation

Question

So I have a problem that asks

decide which ones of the three properties (symmetric, reflexive, transitive) are true for each of the following relations in the set of all positive integers.

the one I'm working on is $m\sim n \iff m\leq n$. Now, the teacher posted solutions saying it is reflexive and transitive, but not symmetric. I am confused because I thought it was symmetric. My reasoning is that while $n$ cannot be less than $m$, the equality still holds, i.e. if $m\leq n$, $n\leq m$ is also true when $n=m$. Am I wrong?

Answer 1

"Symmetric" would mean absolutely every time you have $m \sim n$, you must have $n \sim m$. Notice that, for example, $1 \leq 2$; so $1 \sim 2$, by definition. But $2 \nleq 1$, so $2 \nsim 1$.

Certainly if $n = m$, symmetry holds; but when you're talking about equivalence relations, you don't get to add qualifiers like "if $n = m$".

Answer 2

Symmetric means $\forall m \forall n [m \le n \to n \le m]$, which is clearly false.

What that means is that it needs to work for all $m$ and $n$, not just those that are equal.