CM2010 Explain your reasoning: a) Without using the truth table, prove if the following statements are tautologies or not. Show your reasoning: Fundamentals of Computer Science Assignment, NTU, Singapore

CM2010 Fundamentals of Computer Science: Midterm Coursework

1)  . Explain your reasoning:
a) Without using the truth table, prove if the following statements are tautologies or not. Show your reasoning. [4 marks]
i) 𝒑⋁(𝒑 ⟶ 𝒒)⋁¬𝒒
ii) ((P→q) ∧ ((q∧r)→s)) ∧ (r→(P→s))
b) Without using the truth table, show that the following statement is true: [2 marks]¬(∃x[P(x)∧Q(x)])≡∀x[P(x)→¬Q(x)]
c) Negate the following statement: [2 marks]
∀𝒙:𝑷(𝒙) ∧ [∃𝒙: (𝑸(𝒙) ∧ ¬𝐑(𝐱))]
d) For the following statement, write down a logically equivalent statement which contains no operators other than ¬ and ∨: [2 marks]
𝑷 ∧ (𝑸 → 𝑹)