Let R be a reflexive and transitive relation on X. Show that R intersection R^-1 is an equivalence relation on X.

My proof:

Let R be reflexive and transitive, therefore R (x,x) (y,y) (z,z) (x,y) (y,z)

and R^-1 (x,x) (y,y) (z,z) (y,x) (z,y).

By the intersection rule we have R intersection R^-1 (x,x) (y,y) (z,z).

Is this correct?

How can I show that R intersection R^-1 is an equivalence relation on X?

Thank you.