Help with questions 1,2, and 3

1. Write a proof of the following predicate logic theorem (predicates P and Q are not defined).

1. Write a proof of the following predicate logic theorem (predicates P and Q are not defined). The DoD is the set of integers. (2) Given: (V(x) Q(x) (B) P(y)), 3(x) Q(x) Prove: P(4) Construct a contrapositive proof of the following theorem. Indicate your assumptions and conclusion clearly. (2) 2. Theorem: If you have a bag containing only red balls and white balls, then when three balls are chosen from the bag, at least two of them will have the same color. 3. Let P(x) be the proposition: Given positive integer x, for all positive real numbers a and b, (a+ b)*x >- ax + bAx Prove P(0) using a vacuous proof. (1)