Mathematics for Computer Science

    Question 1. [4 marks] (a) Let p and q be statements. Write down a compound statement that uses only {∧, ∨, ∼} (not necessarily all of them) and is true only when both p and q have the same truth value. Justify your answer using a truth table. (b) Is ∼ q ⇒ q ∧ (p ∨ ∼ q) a tautology, fallacy or contingent statement? Justify your answer. Question 2. [2 marks] Prove that for every natural number n, the number 4 + n + n 2 is not prime. Question 3. [3 marks] Using the substitution and logical equivalence laws, prove the following equivalence. Do not use a truth table. p ↔ q ≡ (∼ q ∨ p) ∧ (∼ p ∨ q) Question 4. [2 marks] Prove or disprove the validity of the following argument. If Scott Morrison is re-elected, the majority of Australians will not be happy. Scott Morrison is not re-elected. Therefore, most Australians are happy.