I'm reading Invitation to Discrete Mathematics (2nd edition) by Matousek and Nesetril. Page 41, problem #2 asks:
Prove that a relation $R$ on a set $X$ satisfies $R ◦ R^{-1} = ∆X$ if and only if $R$ is reflexive and antisymmetric.
I don't see the truth in this statement. A counterexample can be the relation $∆X\bigcup(3,4)$ with inverse $∆X\bigcup(4,3)$. It is reflexive and antisymmetric. But $R ◦ R^{-1} = ∆X\bigcup(4,3)\bigcup(3,4)$ since $3R4, 4R^{-1}4\implies 3(R ◦ R^{-1})4$ and $4R4, 4R^{-1}3\implies 4(R ◦ R^{-1})3$. What am I missing?
