Prove that if the relation (R) is symmetric and antisymmetric on the set X, there exists a Y subset of X such that R is the = relation on Y.

Question

Prove that for any relation R on a set X that is both symmetric and antisymmetric, there is subset Y \subseteq X for which R is the relation = on Y.

I will tell you what I know. I know that relations are sets and that I can show two relations are equal by showing they are subsets of each other. I also am itching to say that Y is the domain of R. But I have no clue how to start.

Answer 1

Yes, $Y$ is the domain of $R$.

HINT: Show that if $R$ is symmetric and antisymmetric then every pair in $R$ has the form $(a,a)$.

Let me expand a little bit on that hint with a series of steps. Let us denote by $E_A$ the relation $\{(a,a)\mid a\in A\}$. Denote by $Y$ the domain of $R$.

1. Show that if $(x,y)\in R$ then $x=y$, and conclude that $R\subseteq E_Y$.
2. Show that if $x\in Y$, then $(x,x)\in R$.
3. Conclude equality.