Suppose that two binary operations, denoted by + and * are defined on a nonempty set S, and that the
following conditions are satisfied for all x, y, and z in S:
(1) x+y and x * y are in S
(2) x+(y+z) = (x+y)+z and x*(y*z) = (x*y)*z
(3) x+y = y+x
Also, for each x in S and for each positive integer n, the elements nx and n x are defined recursively as x^n are defined recursively as follos:
1x = x^1 = x and
If kx = x^k have been defined, then (k+1)x = kx + x and x^(k+1) = x^k * x.
Which of the following must be true?
I. (x*y)^n = x^n * y^n for all x and y in S and for each positive integers n.
II. n(x+y) = nx ny for all x and y in S and for each positive integer n.
III. x^m * x^n = x^(m+n) for each x in S and for all positive integers m and n.
(A) I only
(B) II only
(C) III only
(D) II and III only
(E) I, II, and III