MATH111 Numbers and Relations

End-of-module test, November 2015

The test will last 40 minutes. Attempt all questions. Write your tutor’s name on the first sheet and your own name on each sheet. The number in brackets at the end of each question is the number of marks available; the total score is 50.

1. 1.

   Let $p$, $q$ and $r$ be statement variables. Find the truth tables of the compound statements

   $$p\text{ or }(q\,\&\,r)$$

   and

   $$(p\text{ or }q)\,\&\,(p\text{ or }r),$$

   and hence decide whether these two statements are logically equivalent.

   [8]

2. 2.

   For each of the two numbers $4185$ and $4295$, either express it as an integral linear combination of $945$ and $225$, or explain why this is impossible. [7]

3. 3.

   Find the smallest positive solution to the congruence

   $$35\equiv 20\bmod 55.$$

   [10]

4. 4.

   Define a relation $\sim$ on the set $\mathbb{Z}$ by stipulating that

   $$x\sim y\text{ if }x+7y\text{ is divisible by }4$$

   Decide whether this relation is reflexive, symmetric and transitive, and justify your answers. [10]

5. 5.
   • (i)

     Prove that $\sqrt{2}$ is irrational. [10]

   • (ii)

     Deduce that

     $$(\forall x\in\mathbb{R})[(\sqrt{2}-x\not\in\mathbb{Q})\text{ or }(\sqrt{2}+x\not\in\mathbb{Q})].$$

     [5]

End of Test