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We tell four tales of De Morgan.

In each case we have something that looks like AND, something that looks like OR, and something that looks like NOT.

## Sets

### The Collection of Objects

Consider a universe of discourse/universal set/ambient set  $U$. When talking about people this might be the collection of all people. When talking about natural numbers this might be the set $\mathbb{N}$. When talking about real numbers this might be the set $\mathbb{R}$. When talking about curves it might be the set of subsets of the plane, $\mathcal{P}(\mathbb{R}\times \mathbb{R})$, etc.

The collection of objects in this case is the set of subsets of $U$, denoted $\mathcal{P}(U)$.

Suppose, for the purposes of illustration, that

$U=\{1,2,3,4,5\}$.

Consider the subsets $A=\{2,3,5\}$, and $B=\{1,3,5\}$ .

in the obvious way.

### AND

Note that two objects are contained both in $A$ AND in $B$. We call the set of such objects the intersection of $A$ AND $B$, $A\cap B$:

$A\cap B=\{3,5\}$.

We can represent the ambient set $U$, as well as the sets $A$ and $B$ — and the fact that they intersect — using a Venn Diagram:

We can demonstrate for a general $A$ and $B$ ‘where’ the intersection is:

• The first piece of advice is to read questions carefully. Don’t glance at a question and go off writing: take a moment to understand what you have been asked to do.
• Don’t use tippex; instead draw a simple line(s) through work that you think is incorrect.
• For equations, check your solution by substituting your solution into the original equation. If your answer is wrong and you know it is wrong: write that on your script.

If you do have time at the end of the exam, go through each of your answers and ask yourself:

2. does my answer make sense? If no, say so, and then try and fix your solution.
3. check your answer (e.g. if you are looking at a general true, look at a special case; substitute your solution into equations; check your answer against a rough estimate; or what a picture is telling you; etc). If your answer is wrong, say so, and then try and fix your solution.

## Student Feedback

You are invited to give your feedback on my teaching and this module here.

## Week 12

We will finish our study of Graph Theory by looking at Eulerian graphsHamiltonian graphs, and Dirac’s Theorem.

We had the test on Tuesday.

On Friday we had a look at this graph, a Chapter 4 question:

## Student Feedback

You are invited to give your feedback on my teaching and this module here.

## Test 2

Test 2, worth 15% of your final grade, based on Chapter 3: Algebra, and will take place on Tuesday 3 December in the usual lecture venue of D160.

The sample is to give you an idea of the length of the test. You know from Test 1 the layout (i.e. you write your answers on the paper). You will be allowed use a calculator for all questions.

I strongly advise you that, for those who might have done poorly, or not particularly well, in Test 1, attending tutorials alone will not be sufficient preparation for this test, and you will have to devote extra time outside classes to study aka do exercises.

If you go into Canvas, and go into MATH6055 and the ‘Algebra’ unit, you might see online practise questions for Test 2.

## Week 11

On Monday we half-finished the Examples of Functions (we will finish this off on Friday) mini-chapter before starting the final chapter, the easy chapter, on Network (Graph) Theory.

## Week 12

We will finish our study of Graph Theory by looking at Eulerian graphsHamiltonian graphs, and Dirac’s Theorem.

The Test is on Tuesday.

## Transposition Project – Survey

With regard to TRANSPOSITION (pages 86-97 of the manual), please fill out this survey.

## Transposition Project – Part II

October 14 you took a quiz as part of the Transposition Project that the Mathematics Department is undertaking in an effort to improve our teaching.

You will have another 15 minute quiz on Monday.

If you do not have an internet ready device, or did not do the first quiz, you may leave class early.

Thank you again for your participation.

## Test 2

Test 2, worth 15% of your final grade, based on Chapter 3: Algebra, and will take place on Tuesday 3 December in the usual lecture venue of D160.

The sample is to give you an idea of the length of the test. You know from Test 1 the layout (i.e. you write your answers on the paper). You will be allowed use a calculator for all questions.

I strongly advise you that, for those who might have done poorly, or not particularly well, in Test 1, attending tutorials alone will not be sufficient preparation for this test, and you will have to devote extra time outside classes to study aka do exercises.

## Exercises on Canvas

If you go into Canvas, and go into MATH6055 and the ‘Algebra’ unit, you might see online practise questions for Test 2.

## Week 10

We worked with logarithms and started a quick look at Examples of Functions (which we needed Algebra to talk about).

## Week 11

On Monday we will finish the Examples of Functions mini-chapter before starting the final chapter, the easy chapter, on Network (Graph) Theory.

## Test 2

Test 2, worth 15% of your final grade, based on Chapter 3: Algebra, and will take place on Tuesday 3 December in the usual lecture venue of D160.

The sample is to give you an idea of the length of the test. You know from Test 1 the layout (i.e. you write your answers on the paper). You will be allowed use a calculator for all questions.

I strongly advise you that, for those who might have done poorly, or not particularly well, in Test 1, attending tutorials alone will not be sufficient preparation for this test, and you will have to devote extra time outside classes to study aka do exercises.

## Week 9

In Week 9 we finished talking about equations (and quadratic equations) and began studying exponents. We saw at the very end of Friday’s lecture that if we define a function:

$2^{\Box}:\mathbb{R}\rightarrow \mathbb{R}^+$,

by $2^{\Box}(x)=2^x$, then we define $\log_2:\mathbb{R}^+\rightarrow \mathbb{R}$ as an inverse function:

$\log_2:=(2^{\Box})^{-1}$.

## Week 10

We will introduce more and study the properties of uses of logarithms.

## Week 9

In Week 9 we will finish talking about equations (and quadratic equations) and begin studying exponents.

If you have time to look at MATH6055 over the Reading Break, may I suggest that you look at the Functions Exercises:

P. 61, Q. 1-11, 13
P. 54, Q. 1-4
P. 46, Q. 1-5
P. 41, Q. 1-2, 3 (a), 4, 5 (a)
If you have any questions please email me, perhaps with a photo of your work.

## Catch Up Lecture

13:00 this coming Wednesday 6 November, in B217.

## Week 7

In Week 7 we started delving more into algebra

## Week 8

In Week 8 we will start talking about equations.

## Catch Up Lecture

We will not have MATH6055 class next Friday, 09:00, 25 October.

I would like to catch up on this after the reading break. Please click here and fill out your preference between the times. I know ye are all free at this times and it is just a case of picking the most popular time.

## Week 6

We finished Chapter 2 with a Concept MCQ on Functions Theory, and moved onto Algebra.

## Week 7

In Week 7 we started delving more into algebra and started talking about equations.

## Test 1

Test 1, worth 15% of your final grade, will take place in the usual D160 lecture venue at 10:00 on Tuesday 15 October.

You can find a sample here.

Test 1 will cover the contents of Chapter 1 and the sample will give an idea of the layout and length of the test.

The following types of questions are examinable:

• P.22, Q. 1, 3, 4, 6-7, 9, 13-14
• P.28, Q. 1-4
• P.32, Q. 2
• P. 37, Q.1-3

Questions like those exercises not listed here will not appear on your exam paper but are still useful to help your learning and understanding. For example, working with truth tables should help your understanding of the Laws of Sets.

Any questions at all, you can email me, perhaps with a screenshot of your attempt.

You will want to be familiar with all the concepts in the Chapter Summary, P. 38.

## Transposition Project

On Monday you will be sent a 20 minute quiz that you will take on a mobile internet device — such as your mobile phone — during Monday’s lecture. You will have another quiz again after we finish studying algebra.

These quizzes do not affect your MATH6055 grade but are part of a larger project the Mathematics Department is undertaking in an effort to improve our teaching.

If you do not have an internet ready device you may leave class early.

## Week 5

In Week 5 we continued our study of functions and their properties. We considered the composition of functions — and inverse functions.

## Week 6

We should finish Chapter 2 and move onto Algebra.

## Test 1

Test 1, worth 15% of your final grade, will take place in the usual D160 lecture venue at 10:00 on Tuesday 15 October.

You can find a sample here.

Test 1 will cover the contents of Chapter 1 and the sample will give an idea of the layout and length of the test.

The following types of questions are examinable:

• P.22, Q. 1, 3, 4, 6-7, 9, 13-14
• P.28, Q. 1-4
• P.32, Q. 2
• P. 37, Q.1-3

Questions like those exercises not listed here will not appear on your exam paper but are still useful to help your learning and understanding. For example, working with truth tables should help your understanding of the Laws of Sets.

## Week 4

We finished Chapter 1 by looking more at relations and their properties. We started our study of Chapter 2: Functions Theory. A function is a special type of relation.

## Week 5

In Week 5 we will continue our study of functions and their properties.