It is a known result that if V is a vector space and m(x,y) is a metric on V, then m induces a norm on V defined by ||x|| = m(x,0) where x, y are elements of V only if both

(1) m(x,y) = m(x+a,y+a) and 2) m(alpha*x,alpha*y) = |alpha|*m(x,y).

Since the problem doesn't quality the vector space, I assume that V = R (the real numbers).

A) e(x,y) does induce a norm on R because both (1) and (2) are satisfied: