1.2 Properties of Groups

Recall the definition of a group:

A set G is “upgraded” into a group if it satisfied the following axioms under one binary operation (*) :

  1. Closure: \forall x, y \in  G, x*y \in G
  2. Associativity: \forall x, y, z \in G, (x*y)*z = x*(y*z)
  3.  Identity: \exists e \in G,  \text{ called the identity element  such that } \forall x \in G, x*e = e*x = x
  4. Inverse:  \exists y \in G, \text{ called the inverse of x, with } x*y = y*x = e  \forall x \in G

An Abelian group is a group that is follows the axioms 1 – 4 with the addition of one property: 

  1. Commutativity: \forall  x, y \in G,  x*y = y*x

 

In addition to the axioms, the following properties of groups are important to note:

  1. Uniqueness of the identity element
  2. Uniqueness of the inverse element
  3. Cancellation law
  4. Inverse property (extended)

Uniqueness of an element in mathematics means there exists only one such element with that property. We prove uniqueness by making an assumption that there are two elements in the set that satisfy the property, and show that if such a situation holds, then the two elements must be equal!

We use * to denote the binary operation between elements and “QED” to signal the end of the proof.

The remainder of the post aims to go through the proofs of these properties!…

By | October 28th, 2018|Uncategorized|0 Comments

1.1 Groups Introduction

Binary operations are operations such as addition, subtraction, multiplication, division, modulus etc. that are applied to two quantities.

example 1: 2+5 is an example of an expression with addition as the binary operation

example 2: Let f and g be functions defined on sets A to B. Then the composition of the functions \text{ f(g(x)) } is a binary operation

We will use * to denote an arbitrary (general) binary operation.

A set G is “upgraded” into a group if it satisfied the following axioms under one binary operation (*) :

  1. Closure: \forall x, y \in  G, x*y \in G
  2. Associativity: \forall x, y, z \in G, (x*y)*z = x*(y*z)
  3.  Identity: \exists e \in G,  \text{ called the identity element  such that } \forall x \in G, x*e = e*x = x
  4. Inverse:  \exists y \in G, \text{is called an inverse element of } x \in G \text{ with } x*y = y*x = e

An Abelian group is a group that is follows the axioms 1 – 4 with the addition of one property: 

  1. Commutativity: \forall  x, y \in G,  x*y = y*x

For the remainder of this post, we will explore these axioms and look at some examples

Closure: \forall x, y \in  G, x*y \in G

This means we can take any elements in the set G and perform the operation defined by * and the result will also be an element in the group.…

By | October 28th, 2018|Uncategorized|0 Comments

Prove that for every positive integer n, 9^n – 8n -1 is divisible by 64.

Prove that for every positive integer n, 9^n-8n-1 is divisible by 64.

This question screams proof by induction, so we start with the base case, which in this case is n=1:

9^1-8-1 which is indeed divisible by 64.

Now, let’s assume that it holds true for some positive integer n=k. ie:

9^k-8k-1=64p for p\in\mathbb{Z}.

Now let’s see how we can use this to prove that the statement holds true for n=k+1. For n=k+1 we have:

9^(k+1)-8(k+1)-1=9(9^k)-8k-8-1=9(9^k-8k-1)+64k

where we have manipulated the expression to contain the left hand side of the inductive hypothesis. Thereby, plugging in the inductive hypothesis, we get:

9^(k+1)-8(k+1)-1=9(9^k)-8k-8-1=9(64p)+64k=64(9p+k)

but clearly 9p+k is an integer, so this is divisible by 64 and thus the statement holds true for n=k+1, thus it holds true for all positive integers k

How clear is this post?
By | October 24th, 2018|Courses, First year, MAM1000, Uncategorized, Undergraduate|0 Comments

A tricky complex numbers problem

The question is as follows:

If \frac{\pi}{6}\in arg(z+a) and \frac{2\pi}{3}\in arg(z-a) and a\in \mathbb{R}, find z.

So, let’s think about the information given and what we are trying to find. We want to find the complex number z which satisfies this slightly strange set of constraints, and the constraints are given in terms of z and a. So, by the looks of things, the answer will depend on a and so the final expression should be a function of a.

Now let’s explore the constraints. In fact, let’s simply take z+a and z-a as two complex numbers, but importantly two numbers which differ only by a real number 2a, so wherever they lie in the complex plane, they have the same imaginary part and differ only by an real part.

Now, the constraints are about the arguments of the two complex numbers. It doesn’t tell us anything about the magnitude of the numbers, so all the information tells us is the direction are in relation to the origin.…

By | October 24th, 2018|Courses, First year, MAM1000, Uncategorized, Undergraduate|1 Comment

Using polynomials to solve differential equations

One of the aims of MAM1000W isn’t just to teach you individual mathematical topics, but over time to allow you to see the links between these subjects. Sometimes we do this explicitly, and sometimes you should notice the connections yourself simply by seeing one topic pop up in the middle of another. As I’ve written before, so much of it is about noticing patterns.

Today in class I gave a differential equation which wouldn’t be solvable by any of the methods we have looked at.

y''(x)+\cos(x)y'(x)+e^xy(x)=x^2

This is second order linear but its coefficients are not constant. We don’t have any way in to solve this. We actually wanted to solve this with the initial conditions y(0)=1 and y'(0)=-1.

Actually, that’s a lie. We didn’t want to solve it, but we wanted to get an approximation for the solution close to x=0. This is like saying: OK, so we have a differential equation for population dynamics, or climate change, or the heating of an object, and we don’t worry too much about the very far future, but we want to know what it’s going to do in the short term.…

By | October 15th, 2018|Uncategorized|1 Comment

The Diagnostic Mathematics Information for Student Retention and Success (DMISRS) Project

Presentation by Robert Prince, UCT at the Teaching and Learning of Mathematics Communities of Practice meeting at UJ, 29 – 30 August 2018

robert prince

The Diagnostic Mathematics Information for Student Retention and Success (DMISRS) Project

The problem: Only 27% of students entering full-time university in 2006 graduated in minimum time.

40% leave higher education.

41% of engineering and 48% of science 2006 entrants graduated in 5 years.

 

Educational diagnostic testing is assessment before instruction begins.

DMISRS – a collaboration by mathematicians to improve graduation rates.

  • Make use of NBT data to inform students and lecturers about what areas of weakness and strength are.
  • Share practices, leverage best practices.
  • Extend the reach of academics beyond a physical classroom.
  • Create supportive environments for maths learning [maybe using positive psychology]. What kinds of things will make our classrooms more welcoming to students.

Objectives

  1. Get more institutions on board to collect diagnostic information in the same style/language.
By | September 6th, 2018|Uncategorized|0 Comments

Future Planning of the USAf Teaching and Learning of Mathematics Community of Practice

Professor Rajendran Govender from the University of the Western Cape presented the objectives and future plans of the Universities South Africa (USAf) Teaching and Learning of Mathematics Community of Practice (TLM CoP) at the 2-day meeting at the University of Johannesburg, 29 – 30 August 2018

R Govender

  • In accordance with the principles guiding a community of practice, the TLM CoP provides an opportunity for academics and relevant other university staff members to collaborate, network and share knowledge on issues of common interest or concern.
  • The objectives of the TLM CoP are to promote and strengthen the teaching and learning of Mathematics in public universities in South Africa by:

1.Developing and recommending strategies for the sector to ensure improved access and success in the teaching and learning of Mathematics, thus contributing directly to the transformation needs of South Africa. (session dedicated to this at the next meeting)

2.Providing a shared, common platform from which successful initiatives may be disseminated.…

By | September 6th, 2018|Uncategorized|0 Comments

Radically transforming mathematics learning experiences: Lessons from the Carnegie Math Pathways

Siyaphumelela Conference 2017, The Wanders Club, Johannesburg

Andre Freedman, Capital Community College

Bernadine Chuck Fong, Carnegie Math PathwaysWhatsApp Image 2017-06-27 at 19.00.43 WhatsApp Image 2017-06-27 at 19.00.23 WhatsApp Image 2017-06-27 at 19.00.15 (1) WhatsApp Image 2017-06-27 at 19.00.15 WhatsApp Image 2017-06-27 at 18.59.54 WhatsApp Image 2017-06-27 at 18.57.16 (1) WhatsApp Image 2017-06-27 at 18.57.16 WhatsApp Image 2017-06-27 at 18.38.22 WhatsApp Image 2017-06-27 at 18.37.57 WhatsApp Image 2017-06-27 at 18.37.41 Andre Freedman and Bernadine Chuck Fong

Workshop goals:

  • Learn about the design, goals, implementations of Carnegie Math Pathways
  • Experience Pathways lessons
  • Engage in design tasks to improve student success in maths and college
  • Engage in conversations about professional learning to address issues and concerns that are specific to local contexts

Faculty had to learn new ways to teaching maths, there had to be ‘buy in’ for it to be successful.

How to radically transform outcomes for all mathematics students?

  • Completion
  • persistence
  • quality of learning (e.g. students can explain what a function is years after taking a maths course)
  • identities of learning (students see themselves as someone who can do maths).

Make maths a Gateway not a Gatekeeper.

How?

  • Acceleration: Rather not have 3 developmental courses (pre-algebra, algebra1, 2) before taking the required course
  • Problem-centered curriculum
  • Student-focused, collaborative pedagogy
  • ‘Productive Persistence’ interventions / practices to give students belief that they can succeed
  • Language and literacy supports
  • Train faculty so they can feel comfortable about the new approach
  • Use networking to support staff to get running and sustain change
  • Keep cohort together for 2 semesters in classes of 30 – 40, or as one institution did teach the 2 semesters in one term (quarter of a year) with about 5 hours a day and one other course -success rate was very high at 78%.
By | September 5th, 2018|Uncategorized|0 Comments

Shuttleworth Postgraduate Scholarship Programme

ShuttleworthPostGradScholarship_UCT

How clear is this post?
By | August 23rd, 2018|Uncategorized|0 Comments

Is MAM1000W Making You Anxious?

Hello, my name is Jeremy :-)

I am new to the MAM1000W team of tutors – if you want to read more about my background you can take a look at my bio in the MAM1000W document on Vula. In short, I returned to UCT last year to do my second undergraduate degree, a BSc in Applied Maths and Computer Science, at 25 years old, after not doing any maths for seven years. In the beginning, I found MAM1000W really hard; the pace of the content and the tutorials made me anxious and when test one came around I scored 50%. More anxiety. Luckily I have a great support system (inside and outside the Math department) and with some good advice and determination, I was able to figure out a new, way of studying and managing my time that worked better for me. When it came time for test 2, despite being super stressed out, I scored 81%.…

By | July 22nd, 2018|Uncategorized|1 Comment