Please could anyone confirm the following:
This concerns the relation as defined below on the complex numbers
z~w if Re z = Re w
This relation is reflexive, since the Re z = Re w, so z~w.
Let z,w be an element of a complex number, and suppose z~w. Then Re z = Re w. Hence Re w = Re z, so w~z. Thus the relation is symmetric.
Let z,w,v be an element of a complex number, and suppose that z~w and w~v. Then Re z = Re w and Re w = Re v. It follows that Re z = re v, so z~v. Thus the relation is transitive.
Hence this relation is an equivalence relation.