Given $R = \{(1, 4), (2, 1), (2, 2), (2, 4), (3, 1), (3, 2), (4, 3), (5, 5) \}$ over $A = \{1, 2, 3, 4, 5\}$, an assignment is asking me to prove that $R^{-1} \circ R$ is NOT an equivalence relation.
I'm having a lot of trouble with this because I'm pretty much convinced that $R^{-1} \circ R$ IS an equivalence relation.
$R^{-1} \circ R = \{ (1,1),(1,2),(1,4),(2,1),(2,2),(2,4),(3,3),(4,1),(4,2),(4,4),(5,5) \}$
$R^{-1} \circ R$ is reflexive, symmetric and transitive. Therefore, it is an equivalence relation.
Did I calculate $R^{-1} \circ R$ wrong, or is there something wrong with my understanding of the concept of an equivalence relation? Or maybe there's a mistake in the assignment itself. What is happening here?