Let X be a set. A predicate, P(z), on X is a statement whose truth value depends on the choice of r X. For example, if X-R and P(z) is the predicate z> 1, then P(z) is falso when z(-00, 1] and P(r) is true when (1,0). For another example, if X 111 RXZ and P(x, y) is the predicate z>y+1, then P(x, 3) is false and P(7, 2) is true. Now, let P(z) be a predicate on set X. We write Vr X, P(r) to mean the statement "For all a X, P(r) is true". Similarly, we write z X, P(z) to mean the statement "There exists some TEX, such that P(r) is true", Let R be a ring. Determine (with proof) whether or not the following statements are true or falso: (a) VrER, 3y E R. ry - x. (b) 3r ER, Vy E R. ry-z. (c) Vr E R. Jy R (y0y /0) (d) Vr E R. Jy e R (2-0 Vry /0) Consider R FRE Z with the regular/standard addition + With proof, find all functions : Rx RR such that (R, +,) is a ring.