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## Outlook

### Week 9

*There is only a mini-chapter on examples of (real) functions. You should spend the rest of the time revising Chapters 3 and 4 (i.e. doing exercises). Some of you might have to catch up on the Week 8 lectures.*

### Week 10: 14 to 20 December.

*There will be feedback on submitted exercises Wednesday, Thursday, and Friday.*

*I plan that we will have a number of Zoom tutorials (these are provisional):*

*Tuesday 14:00**Tuesday 15:00 C-X only**Wednesday 12:00 C-Y only**Thursday 12:00*

*There is a small chance that we will also have Monday 14:00 Zoom but will not know until closer to the day.*

### Week 11 – 40% Test 3

*The final, 40% Test 3 on Functions and Algebra (chapters 3-5) will take place (provisionally) at 18:00-19:30 (perhaps with additional upload time) on Monday 21 December. There will be no more feedback on submitted exercises but there might be a 14:00 Zoom before the final Test.*

## 30% Test 2 – Results

I had said that my aim was to have these to you before Wednesday 9 December. However there is a possibility now that the Test 2 results will not be released until after you sit Test 3. This is outside my control (watch this space).

## Week 9

### Lectures

Schedule about an hour and 20 minutes for these:

- Examples of (Real) Functions I (25 minutes)
- Examples of (Real) Functions II [18.30 on is Exercise Hints] (27 minutes)
- Examples of (Real) Functions Summary (3 minutes)

### Exercises

- p.158-163, Q. 1-17 [Hints: [18.30-26.44]]

## Outlook

### Week 8

*As you will see below there is a lot of lecture material. It might be wise to break your learning into Exponents and Logarithms, and spend 3.5 hours on each. Start the clock running on Exponents, after you watch those lectures spend the rest of the 3.5 hours doing exercises on them, and then repeat the trick with logs. *

*That would be ideal but the good news is that we have all but finished the lecture material, there won’t be too many lecture videos in Week 9, there will be none in Week 10, and so there will be scope to catch up. However you will be better off keeping up as you will need time to revise Chapter 3 Functions also.*

*You can always submit whatever work you want to whatever is open on Canvas at the time (but please label properly with page and question number).*

### Week 9

*We will complete the mini-chapter on examples of functions. You will spend the rest of the time revising Chapters 3 and 4 (i.e. doing exercises).*

### Week 10

*There will be feedback on submitted exercises Wednesday, Thursday, and Friday.*

*I plan that we will have a number of Zoom tutorials (these are provisional):*

*Tuesday 14:00**Tuesday 15:00 C-X only**Wednesday 12:00 C-Y only**Thursday 12:00*

*There is a small chance that we will also have Monday 14:00 Zoom but will not know until closer to the day.*

### Week 11

*The final, 40% Test 3 on Functions and Algebra will take place (provisionally) at 18:00-19:30 (perhaps with additional upload time) on Monday 21 December. There will be no more feedback on submitted exercises but there might be a 14:00 Zoom before the final Test.*

## 30% Test 2 – Results

My aim would be to have these for you before Wednesday 9 December.

## Week 8

### Lectures – Exponents

Schedule about an hour and 50 minutes to watch these:

- Exponents/Powers I (25 minutes)
- Exponents/Powers II (25 minutes)
- Exponents/Powers III and Logarithms I [9.35 to 12.40 of Exercise Hints] (24 minutes)

### Exercises – Exponents

- p.99, Q.11 (functions question that requires algebra)
- p.128-129, Q. 1-10 [Hints: [9.35-12.40]]

You are advised to to spend **seven** hours per week on MATH6055. This should comprise of however long it takes to watch the lectures, and then the rest of time should be spent emailing questions, doing exercises, and catch-up/revision.

## 30% Test 2

The second 30% assessment takes place 6 pm Wednesday 25 November, and covers equivalence relations and Chapter 2. Relevant are the Week 3, 4 and 5 announcements on Canvas. Students who have been working on this material are in a good position and should be able to watch the Chapter 3 lectures while those who have not been working (i.e. doing and submitting exercises) are probably going to have to work on the Weeks 3-5 exercises.

Chapters 3 to 5 will be assessed in the final assessment in Week 10/11.

### Zoom for Test 2

There will be two Zoom tutorials for Test 2 in Week 7:

- Monday 23 November, 14:00
- Tuesday 24 November, 14:00

The links have been sent via Canvas Mail. Students are invited to attend one or both tutorials where I will be answering student questions. The tutorials will be recorded.

## Week 7

### Lectures

Schedule about two hours and 15 minutes to watch this hour and a half of lectures.

- Algebra: Intro and Basics (27 minutes)
- Equations I (26 minutes)
- Equations II (27 minutes)
- Equations III (11 minutes)

You are advised to to spend **seven** hours per week on MATH6055. This should comprise of however long it takes to watch the lectures, and then the rest of time should be spent emailing questions, doing exercises, and catch-up/revision.

## 30% Test 2

The second 30% assessment takes place in Week 7 (week of 23 November, provisionally 6 pm Wednesday 25 November), and covers equivalence relations and Chapter 2. Relevant are the Week 3, 4 and 5 announcements on Canvas. Students who have been working on this material are in a good position and should be able to watch the Chapter 3 lectures while those who have not been working (i.e. doing and submitting exercises) are probably going to have to work on the Weeks 3-5 exercises.

Chapters 3 to 5 will be assessed in the final assessment in Week 10/11.

### Zoom for Test 2

There will be two Zoom tutorials for Test 2 in Week 7:

- Monday 23 November, 14:00
- Tuesday 24 November, 14:00

I will send the links for these a little closer to the slots. Students are invited to attend one or both tutorials where I will be answering student questions. The tutorials will be recorded.

## Week 6

### Lectures

Schedule about an three hours to watch this 2 hours and ten minutes of lectures.

- One-to-One, Onto, Bijective I (29 minutes)
- One-to-One, Onto, Bijective II (29 minutes)
- One-to-One, Onto, Bijective: End of Example 9 and Exercise Hints (10 minutes)
- Compositions and Inverses I (29 minutes)
- Compositions and Inverses II (24 minutes)
- Compositions & Inverses Exercise Hints and Functions Summary (9 minutes)

You are advised to to spend **seven** hours per week on MATH6055. This should comprise of however long it takes to watch the lectures, and then the rest of time should be spent emailing questions, doing exercises, and catch-up/revision.

## 30% Assessment 2

The second 30% assessment takes place in Week 7 (week of 23 November), and covers equivalence relations and Chapter 2.

## Week 5

### Lectures

Schedule about an hour and 40 minutes to watch this 68 minutes of lectures.

- Trees, Trails, Cycles (29 minutes)
- Eulerian Trails (12 minutes –
*The original video was 16 minutes long but after 12 minutes is a waste of time and so I deleted it*) - Functions: Definitions and Coordinate Graphs (27 minutes)

You are advised to to spend **seven** hours per week on MATH6055. With the first assessment on Wednesday, much of this time may be taken up revising Chapter 1.

The rest of you time should be spent however long it takes to watch the lectures, and then the rest of time should be spent emailing questions, doing exercises, and catch-up/revision.

## 30% Assessment 1

- When?
**Wednesday Nov 4th**,**6-7 pm**. - Where? This exam will be made available as an
**assignment on Canvas**. - Duration? You will be allotted
**45 minutes for the exam and 15 minutes to upload pictures**of your work via the canvas app, so 1 hour in total. - Topic?
**Sets and Relations**(i.e. Chapter 1 up to but not including equivalence relations) - Marks? This exam will make up
**30%**of your total mark in this module.

If you have any questions in relation to the exam, please contact me.

## Week 4

### Lectures

Schedule about two hours to watch this hour and 20 minutes of lectures.

- Network/Graph Theory: Introduction and Basics (28 minutes)
- Graphs aka Undirected Graphs aka Networks (22 minutes)
- Properties of Graphs (29 minutes)

### Exercises

#### Focus on Assessment 1

If you are revising for the first assessment I recommend:

- If you submitted work on Canvas, read the feedback I gave you. If I recommend trying questions again, etc., do so.
- If all done, go back on previous week’s questions that you did not complete.

You must understand that:

*Any and all submitted work will receive feedback.*

This includes work submitted late, or work emailed to me. It is easiest for me however if work is:

- Submitted on Canvas to Reading Week Exercises
- Properly labelled (page number and question)

You can (carefully) take photos of your work and submit to the Reading Week Exercises those images on Canvas before midnight Monday 2 November for feedback.

Submit the images as a single pdf file. To do this, select all the images in a folder, right-click and press print. It will say something like *How do you want to print your pictures?* Press (Microsoft?) Print to PDF. If possible choose an orientation that has all the images in portrait.

#### Prepared for Assessment 1?

Try

**p.56,***Use your theorem to answer the following. Which of graphs 1-10 have a `route’ that uses every edge exactly once. Find the routes.***p.58,**Q. 1-6**p.60**, Q. 1**p.64,***In Graphs 1-10, give the degree/valency of each vertex*(this might already be done from the p.56 exercise)

Additional/Harder Exercises:

**p.62,***Implement the Task 3 A Algorithm to determine which of graphs 1-10 on the exercise sheet are connected.*

Submit (after Thursday 5 November) before midnight Tuesday 10 November for feedback.

## Week 5

In Week 5 we will complete Chapter 2: Network (Graph) Theory, and begin Chapter 3: Functions.

## Zoom

Unfortunately when I sent the survey about Zoom, I had not yet “submitted” the Canvas feedback on the Week 1 exercises, and so students opinions might not have been fully informed… at the moment some respondents to the survey say they “need Zoom time”. If you want to have your say on this please fill in this survey (Links to an external site.)

This difficult topic will be considered by me again, perhaps after the Reading Break.

## Academic Learning Centre

Have you heard about CIT Maths Online on Canvas? It’s full of helpful Maths and Stats resources, notes, quizzes and videos to help you throughout the whole year.

During this period of remote learning we will be using the **Maths Online** module on Canvas to offer Maths and Stats support to you and answer as many student questions as possible.

**1-to-1** **Maths appointments** will be available through Zoom. There are 32 appointments available per week. Please log on to Maths online to see the most up to date resources and to book a 1-to-1 video call. Find out how to add it to your Canvas dashboard here https://studentengagement.cit.ie/alc/resources.maths.

**Drop-in Maths** is on every Tuesday from 12 – 2. Here you can drop in to ask our lecturer a quick Maths question.

If you have any other question about our remote Maths and Stats supports email Joy and Deirdre at academiclearning@cit.ie.

## Assessment

At the time of writing, the assessment will comprise two 30% assessments and one 40% assessment. Provisionally Weeks 4, 7, and 10.

**seven** hours per week on MATH6055. This should comprise of however long it takes to watch the lectures, and then the rest of time should be spent emailing questions, doing exercises, and catch-up/revision.

## 30% Assessment 1 Week 4 (week of 2 November)

*Provisionally* the first assessment will take place in Week 4. It is provisionally based on everything up to (but not including) Section 1.4.2, but it might make sense (for the purposes of understanding best the earlier part of Section 1.4) to consider Section 1.4.2 part of the assessment.

More definitive information will be posted to Canvas in good time.

## Zoom

In ordinary times, the majority of student learning takes place in tutorials where I am able to give one-to-one help to students. This is not possible at the moment so instead I am giving feedback on student exercises. This is very time consuming but I don’t believe it is possible for me to give you the teaching you need in a Zoom tutorial… but I might be wrong.

Please fill in this survey to tell me your opinion on how the tutorial aspect of my teaching is going.

It is a difficult balancing act. I could just do two Zoom hours per week but I simply don’t believe ye would get the one-to-one help in that way. I would be going way above and beyond my duties to do Zoom AND give weekly feedback on work but I have to hear what ye think.

If you would like to email me directly your thoughts on tutorial time (i.e. you would like some Zoom time and why), you can email me also (although the survey gives you a chance to do this).

## Week 3

### Lectures

Schedule about one and a half hours to watch this one hour of lectures.

- Relations: Symmetric and Transitive (17 minutes)
- Equivalence Relations I (27 minutes)
- Equivalence Relations II and Sets & Relations Summary (17 minutes)

### Exercises

I recommend you spend five and a half hours this week on exercises (as there is so little lecture time). This means:

- If you submitted work on Canvas, read the feedback I gave you. If I recommend trying questions again, etc., do so.
- Once up on your feedback, try the recommended questions (below).
- If all done, go back on previous week’s questions that you did not complete.

You must understand that:

Any and all submitted work will receive feedback.

This includes work submitted late (i.e. Week 2 exercises submitted in Week 3), or work emailed to me. It is easiest for me however if work is:

- Submitted on Canvas
- Properly labelled (page number and question)
- Ideally work that you know is correct (i.e. matches answer in the manual) not be submitted for feedback.

Do/attempt:

**p. 49, Q. 1****p. 50, Q. 2-7****p. 51, Q. 8-9**

**Additional/Harder Exercises:**

**For each of the relations on p.44**: Is the relation transitive? Justify your answer.

You can (carefully) take photos of your work and submit to the Week 3 Exercises those images on Canvas before midnight Tuesday 27 October for feedback.

Submit the images as a single pdf file. To do this, select all the images in a folder, right-click and press print. It will say something like *How do you want to print your pictures?* Press (Microsoft?) Print to PDF. If possible choose an orientation that has all the images in portrait.

## Reading Break

It is currently my understanding that ye will have a Reading Break 26 October to 1 November. You will use this time to complete your Chapter 1 learning (especially *doing exercises*).

## Week 4

In Week 4 we will start Chapter 2: Network (Graph) Theory.

## Academic Learning Centre

Have you heard about CIT Maths Online on Canvas? It’s full of helpful Maths and Stats resources, notes, quizzes and videos to help you throughout the whole year.

During this period of remote learning we will be using the **Maths Online** module on Canvas to offer Maths and Stats support to you and answer as many student questions as possible.

**1-to-1** **Maths appointments** will be available through Zoom. There are 32 appointments available per week. Please log on to Maths online to see the most up to date resources and to book a 1-to-1 video call. Find out how to add it to your Canvas dashboard here https://studentengagement.cit.ie/alc/resources.maths.

**Drop-in Maths** is on every Tuesday from 12 – 2. Here you can drop in to ask our lecturer a quick Maths question.

If you have any other question about our remote Maths and Stats supports email Joy and Deirdre at academiclearning@cit.ie.

## Assessment

At the time of writing, the assessment will comprise two 30% assessments and one 40% assessment. Provisionally Weeks 4, 7, and 10.

## Week 2

You are advised to to spend **seven** hours per week on MATH6055. This should comprise of however long it takes to watch the lectures, and then the rest of time should be spent doing exercises, emailing questions, and submitting work. At the time of writing, the mechanism for submitting work is not yet set up.

### Lecture

Schedule about three hours to watch these two hours of lectures. I recommend about 50% extra time as you will want to pause/rewind.

- Laws of Sets and Illustrations of DeMorgan’s Laws (25 minutes)
- Inclusion/Exclusion Principle; and some difficult notational stuff (14 minutes)
- Cartesian Products (21 minutes)
- Tuples: Passwords and Bit Strings (7 minutes)
- Relations (22 minutes)

Here is the first tranch of notes if you have not purchased nor printed off a manual

- Relations: Examples and Reflexivity (27 minutes)

### Exercises

I recommend spending four hours on this week’s exercises.

I do not at the time of writing have the mechanism in place for submitting work.

You can (carefully) take photos of your work. If possible, convert the images to a single pdf file. To do this, select all the images in a folder, right-click and press print. It will say something like *How do you want to print your pictures?* Press (Microsoft?) Print to PDF. If possible choose an orientation that has all the images in portrait.

Ordinarily I would encourage you to ask questions via email at any time but at present this mechanism is not in place.

Do/attempt:

**p. 32, Q. 10-11****p. 33, Q. 12-14****p. 38, Q. 1-5****p. 39, Q. 6****p. 43, Q. 1-2**

**Additional/Harder Exercises:**

**p.34, Q. 26****p. 39, Q. 7-8****p. 43, Q. 3-6**

## Week 3

In Week 3 we will finish Chapter 1 by looking more at relations and their properties. We will start our study of Chapter 2: Functions. A function is a special type of relation.

## Academic Learning Centre

I would urge anyone having any problems with material that isn’t being addressed in the tutorial communication to use the Academic Learning Centre. If you are a little worried about your maths this semester you need to be aware of this resource. You will get best results if you come to the helpers there ith *specific *questions.

## Assessment

At the time of writing, the assessment schedule has not been finalised, but we are considering two 50% tests.

## Student Resources

Please see the Student Resources tab on the top of this page for information on the Academic Learning Centre, etc.

## Facetime

Some Facetime with me: click here.

## Paternity Leave

I will be on paternity leave until Monday 12 October. My colleague Dr Mark Hartnett (mark.hartnett@cit.ie) will answer any email queries that you have.

## Manuals

The lectures are being delivered via pre-recorded lectures. As you will see, the lectures use a manual that contain all the lecture material, via gaps that are filled in during lectures, and exercises. I tend to use a number of colours during lectures, and pencil, so you might want to consider ordering some of these:

In a sliding scale from best to worst, in my opinion, here are your options for using this manual. There are other options but I cannot recommend them. If you do option one you have all your notes in one place and can follow the lectures as if you were in the classroom.

- Email copy.centre@cit.ie and tell them you want to order a bound copy of
*MATH6055 Manual Winter 2020*. The manuals can be collected from Reprographics beside the Student Centre. Note that this is a cash-free area so you will need to put the appropriate amount of funds on your student card. At the time of writing I do not know the cost but it will be of the order of €13. This seems like a lot of money for a manual but with all the materials (including exercises, summaries, etc) it comes to 165 pages and provides a comprehensive resource for this module. - Print off the manual at home or somewhere else. Click here to find a copy.
- I am going to scan and email the completed slides. You can keep these somewhere for your notes. You could print these or keep digital copies. Here is the first tranch of notes.

## Week 1

You are advised to to spend **seven** hours per week on MATH6055. This should comprise of however long is recommended to watch the lectures, and then the rest of time should be spent doing exercises, emailing questions, and submitting work. At the time of writing, the mechanism for submitting work is not yet set up.

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 . When talking about people this might be the collection of all people. When talking about natural numbers this might be the set . When talking about real numbers this might be the set . When talking about curves it might be the set of subsets of the plane, , etc.

The collection of objects in this case is the set of subsets of , denoted .

Suppose, for the purposes of illustration, that

.

Consider the subsets , and .

in the obvious way.

### AND

Note that two objects are contained both in AND in . We call the set of such objects the *intersection *of AND , :

.

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

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

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