Part VI  Construct a formal proof of validity for each of the following theorems (using CP and/or RAA if and as you wish A)  ~D → (D → B) B) without using exportation [A → (B → C)] ↔ [(A • B) → C]

e) Let A(x), C(x), S(x) and GP(x) be the statements "x is an animal," "x is a cat," "x is small," and "x is a good pet," respectively. Express each of these quantifications in English where the universe consists of all animals. i. Vx (C(x) → A(x)) ii. -3x(C(x) ^ ¬ S(x)) i. Vx (C(x) S(x) A A(x)) iv. Vx (S(x) A A(x)→ GP(x))

1. Let J₁ = {0, 1,2,3} and define f: J₁ J₁ and g: J₁ J₁ by f(x) = (x + 3)² mod 4 and g(x)=x²-3x +- 5 mod 4. Is f = g? Explain your answer.

Use the rules of inference to show that if 3x(M(x) AP(x)) and Vx(K (x) → ¬M(x)) are true, then 3x(P(x) ^ ¬K(x)) is also true, where the domains of all the quantifiers are the same.