$P$ -- partial preorder. $\theta(P)=\{(x, y)\in A^2 | (x, y) \in P \land (x,x) \in P \}.$ $\theta(P)$ is an equivalence relation: can't see symmetry.

Question

Let $P$ be a partial preorder (which is a reflexive and transitive relation) on an arbitrary set $A$. Consider binary relation $\theta(P)=\{(x, y)\in A^2 | (x, y) \in P \land (x,x) \in P \}.$

My textbook says it's easy to see $\theta(P)$ is an equivalence relation on $A$, but I am completely lost on symmetry, can't see how $x\theta(P)y$ implies $y\theta(P)x$. Is it even true? Is there a typo in the book?

Thank you.

Answer 1

There is a typo. It should be $$ \theta(P)=\{(x, y)\in A^2 \mid (x, y) \in P \text{ and }(y,x) \in P \}. $$