Suppose that R and S are relations on a non-empty set A. Prove/disprove that if $R$ and $S$ are both anti-symmetric then $R\setminus S$ is anti-symmetric.
I am pretty sure that this is true but I am having proving it. Here is what I currently have, i am not sure if this is correct:
Suppose that $a,b\in A$ and $(a,b),(b,a)\in R\setminus S$ this means that $(a,b),(b,a)\in R$ and $(a,b),(b,a)\notin S$ from the definition of set difference. Since both $R$ and $S$ are antisymmetric it follows that $a=b$. Therefore $(a,b)\in R\setminus S \land (b,a)\in R\setminus S \implies a=b$ and $R\setminus S$ must be antisymmetric.