1. Let P(X) be the predicate "x is a dragon." Let Q(r) be the predicate "x breathes fire." Let R(x, y) be the predicate "x and y are the same object." Let S be an arbitrary nonempty set (a) (16 pts) Rewrite the following English statements in symbolic no- tation using predicates P, Q, R, universal and existential quanti fiers, and any variables you want. i. There are no dragons in S. ii. Not everything in S is a dragon. iii. There is at least one dragon in S iv. There are at least two dragons in S. v. There is at most one dragon in S. vi. There is at exactly one dragon in S. vii. Dragons in S breathe fire viii. If an element of S breathes fire, it must be a dragon (b) (14 pts) Rewrite the following statements in English iii. (Vx E S)[Q(x) P(x)]