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.