Provide either a proof or a counterexample for each of these statements. (a) For all positive integers x, x + x +41 is a prime. (b) (Vx)(y)(x + y = 0). (Universe of all reals) (c) (Vx)(Vy)(x > 1^y> 0y > x). (Universe of all reals) (d) (e) For integers a, b, c, if a divides bc, then either a divides b or a divides c. For integers a, b, c, and d, if a divides b-c and a divides c - d, then a divides b d. (f) For all positive real numbers x, x - x 0. (g) For all positive real numbers x, 2 > x + 1. (h) (i) For every positive real number x, there is a positive real number y less than x with the property that for all positive real numbers z, yz 2 z. For every positive real number x, there is a positive real number y with the property that if y < x, then for all positive real numbers z, yz 2 z.