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    MATH221 – Mathematics for Computer Science – Autumn 2022

    Assignment One – Due Week 6 Friday 5:00pm

    Student Name: Student Number:

    Tutorial Day & Time:

    Question 1. [4 marks]

    (a) Let p and q be statements. Write down a compound statement that uses only {∧,∨,∼} (not necessarily all ofthem) 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 + n2 is not prime.

    Question 3. [3 marks] Using the substitution and logical equivalence laws, prove the following equivalence. Donot 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.

    Question 5. [4 marks] Prove by mathematical induction that 12 + 32 + 52 + · · ·+ (2n−1)2 = 4n3−n3

    for

    all n ∈ N.

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