Question

Call a relation R “orbital” if xRy and yRz imply zRx. Prove that R is an equivalence relation if and only R is both reflexive and orbital. (Note that this is an “if and only if” statement, which is bi-conditional. So there are actually two different implications to show here.)

This is what I have so far, can you check to see if this is correct?

If R is an equivalence relation then R has to be reflexive, symmetric, and transitive. This means that xRx which is reflexive. Also, xRy and yRx which means that it is symmetric.Then, xRx is also orbital or transitive if xRy and yRz which implies that zRx. This means that R is an equivalence relation. If R is reflexive then xRx. If R is orbital, then xRy and yRz which implies that xRz. This means that R is an equivalence relation.