# MATH221 – Mathematics for Computer Science – Assignment One Solved

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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

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

• assignmentone.zip