## Description

**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 *= 3*q *+ *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*n*^{2 }+*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*

∼ *P*

∴ ∼ *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