Are all relations that are symmetric and anti-symmetric a subset of the reflexive relation?

Question

Simple question, but I can't seem to find a guaranteed answer. A symmetric set contains (a, b) if it contains (b, a), but an anti-symmetric set only contains (a, b) and (b, a) when a = b. It seems to me like the only time a set can be anti-symmetric and symmetric is if it is a subset of the reflexive set. Is this true?

Answer 1

Let $R$ be a relation on $A$ such that it's both symmetric and anti-symmetric.

Let $(x,y)\in R$. Then by symmetry, $(y,x)\in R$ and by anti-symmetry, $x=y$.