Reference no: EM132248122

Assignment Questions -  

Answer all the questions with full detail and explanation.

Q1. Use the Hilbert-style to prove the following:

a. A, B |- A ≡ B

b. A, ¬A |- ⊥ (Note: Do not prove this via the cut rule).

C A → B |- C ν A → C ν B

Q2. Use the Equational-style to prove the following:

a. |- A ν B ≡ A ν ¬B ≡ A

b. |- A ν A Λ B ≡ A

c. |- A ν (B → A) ≡ B → A

d. A → B ≡ ¬B → ¬A

Q3. Use the Deduction Theorem to prove the following:

a. |- (A → B) → (¬A → ¬B) → B

b. |- ((A → B) → A) → A

Q4. Prove (ii) of 2.4.23 as a consequence of (i); i.e., using (i) as a hypothesis.

2.4.23 Theorem. (Distributivity: ν over Λ and Λ over ν)

(i) |- A ν B Λ C ≡ (A ν B) Λ (A ν C)

and

(ii) |- A Λ (B ν C) ≡ A Λ B ν A Λ C