Let S be a non-empty set of real numbers which is bounded above, and suppose that x = sup(S). - (a) Prove that for all € > 0 there exists s € S so that s ≤ (x − €, x].

5. Prove the statement: Let X and Y be two finite sets with |X|=m and |Y|=n. If there is a one-to-one function f: X → Y, then m≤n.

1. (a) (b) (c) (d) Prove or disprove that, for any universal set U and predicates P and Q, [3x = U, P(x) ^ Q(x)] → [3r EU, P(x)) ^ (3x = U, Q(x))] Prove or disprove that, for any universal set U and predicates P and Q, [3r EU, P(x)) ^ (3xU, Q(x))] → [r U, P(x) ^ Q(x)] Prove or disprove that, for any universal set U and predicate P [3r € U, P(x)] → [Vr € U, P(x)] Prove or disprove that, for any universal set U and predicate P [VxU, P(x)] → [3r € U, P(x)]

Prove that if A is a nonempty set which is bounded below by 3 then B = {x ∈ R | ∃a ∈ A, b = (1/a)} is bounded above by (1/3).

Z is the set of integers, R is the set of real numbers. 2) For each of the following functions prove whether they arei) one to one;   ii) onto.If they are not one to one or onto, show why.a) f: Z -> Z, f(n) = n - 6;b) f: Z -> Z, f(n) = n3;c) f: Z -> Z, f(n) = n2 + n;