Reflexive property of relations?

Question

If a relation is both symmetric and anti-symmetric, isn't it always reflexive? I am supposed to consider a set, S, where some Relation A is both symmetric and anti-symmetric. I thought that anti-symmetric means that if (x, y) is in the relation, then (y, x) cannot be in it unless x = y. So, wouldn't it hold that it is reflexive?

Answer 1

Reflexive means for all elements must be related to themselves but that other elements may (or may not) be related to each other.

Both Symmetric and anti-symmetric means elements can only be related to the themselves (thought they don't need to be) and cant be related to any other elements[1].

So a relation that is symmetric and anti-symmetric need not be reflexive if not all elements are related to themselves. (This would mean those elements are not related to anything.)

Example: If you have a set $S=\{1,2\}$ and $1 \sim 1$ but $2$ is not related to anything that is symmetric. (As every case $a \sim b$ then $a =b = 1$ and $b \sim 1$.) And antismmetric. (As $a\sim b$ and $b\sim a$ means $a = b =1$). But it is not reflexive as $2\not \sim 2$.

[1]. Symmetric means if $a \sim b$ then $b \sim a$. Anti-symmetry means if $a \sim b$ and $b\sim a$ then $a =b$. If a relation is both then $a \sim b \implies b\sim a \implies a = b$.