Question 1. [2 Marks] Consider your student number, n to be a natural number. Find natural numbers q,r with 0 ≤ r < 3 such that n = 3q + r.
Question 2. [2 Marks] Using the value of r computed in Question 1 above answer only part (r) of this question:
- Let P, Q be statements. Write down a compound statement that is true when one and only one of P or Q is false. Justify your answer using a truth table.
- Is ∼ Q ⇒ P ∨ (P∧ ∼ Q) a tautology, a contradiction or a contingent statement? Justify your answer.
- Prove, using cases, that for every natural number n ≥ 1 the expression n2 + n +4 is not a prime number.
Question 3. [2 Marks] Use a truth table to show that the following is a valid argument.
P ⇒ Q
∴ ∼ Q.
Question 4. [4 Marks] In this induction question, full marks will only be awarded for writing out a full argument, like those in the examples from lectures. That is, make it clear which step you’re doing, and write out what Claim k and Claim k + 1 are, then wrap up the argument with a concluding sentence (“Therefore, by induction,
Prove by mathematical induction that for all n ∈ N.
MATH221 – Mathematics for Computer Science
Assignment One, Autumn 2017