Lecture 2: Sets

List of the things I learnt today


  • Definition of a set
  • Different ways of representing a set
  • Different kinds of sets
  • Intervals
  • Combinations of sets
  • Set notation in functions

 

Definition of a set

A set is a collection of objects

Different ways of representing a set

Sets of objects are usually denoted by uppercase letters e.g.

let \mathit{A} be the set of all odd numbers.The objects contained in a set are called elements which are denoted by lowercase letters e.g. \mathit{a} is an element of \mathit{A}.

A set can be represented in two ways:

1.List

\mathit{A}=\{a, b, c, d\}

The set above is an finite list, you can count the number of elements.

\mathit{B}=\{a, b, c, d,... t\}

Is also a finite list. It’s important to present enough elements to produce a sequence if the use of ellipses is implemented for shorthand purposes.

\mathit{C}=\{1, 2, 3, 5, 8,... \}

This is an infinite set, you cannot count the number of elements in the set.

2. Set Builder Notation

\mathit{D}=\{e \in \mathbb{R} : \mathit{C} (e) \}

In the example above e is an element in the set \mathbb{R} where \mathit{C} (e) is a characteristic of the elements which are specific to set \mathit{D} or you can replace “:” with”|”.…

By | February 17th, 2016|Uncategorized|3 Comments

Lecture 1: Ways to represent a function.

So today we basically learnt about different ways to represent a function and we defined what a function is in detail.

First things first, what is a function?

A function is a rule that assigns each element x ( x being any independent variable ) in a set A exactly one element, f(x) ( f(x) being a dependent variable ) in a set B.

From that definition of a function we can now distinguish the different types of ways to represent a function.

I only know four ways, maybe there’s more… idk

  1. algebraically, with the use of an equation
  2. graphs
  3. tables
  4. or just in words

So we also learnt how to test if a graph represents a function. To do that you have to use the vertical line test. which means that anywhere within the domain of the graph if an x value has more than one f(x) value assigned to it then that graph does not represent a function.…

By | February 15th, 2016|Uncategorized|2 Comments

Gravitational waves: will the global south provide the next pulse of gravity research?

Gravitational waves: will the global south provide the next pulse of gravity research?

Amanda Weltman, University of Cape Town and Jeff Murugan, University of Cape Town

A little over a century ago, on 25 November 1915, Albert Einstein published a paper entitled “Die Feldgleichungen der Gravitation”. Its contents would change the world forever.

Like any good scientific theory, Einstein’s General Relativity not only explained the shortcomings of its predecessor, in this case Newtonian gravity, it also made predictions of new and unexpected phenomena. These included the bending of light by massive objects, the existence of black holes, the slowing down of time in strong gravitational fields and the very framework for the cosmology of the universe. All of these have withstood a century of intense scrutiny. But for 100 years one particular prediction in Einstein’s theory of Gravity eluded the most ingenious testing.

That changed on 11 February 2016 with the news that gravitational waves have been discovered.…

By | February 14th, 2016|Uncategorized|0 Comments

The Equation of Everything

Salahdin Daouairi unlocks a numerical model puzzle based on a finite discrete system that is key to understanding and interpreting physical laws of the universe. Read more here.

How clear is this post?
By | February 9th, 2016|Uncategorized|3 Comments

You’re (probably) a Bayesian – whether you like it or not!

Statisticians have long been separated into two camps as to how they philosophically interpret their trade. These schools of thought are usually called Frequentists and Bayesians.

Frequentists believe that a probability, p\in[0~ 1] , associated with a specific possible outcome of an observable occurrence or process, is simply telling you that, could you observe this occurrence (or process) infinitely many times, the fraction of such observations that would yield that specific outcome is p . Using the age-old coin toss example: tossing the coin is the occurrence or process and recording a Heads or Tails are the two observation. The number 0.5 \left(P(\text{Tails})=0.5=P(\text{Heads})\right) tells a Frequentist that, in the pursuit of infinitely many coin tosses, the ratio of Heads recorded to the number of tosses performed asymptotically approaches 0.5. And that’s all! The value should not be interpreted as the most likely outcome for the next observation or sample taken from the process (though I’ve always wondered how a Frequentist would gamble…).…

By | February 3rd, 2016|English, Fun, Level: Simple, Uncategorized|0 Comments

NFL Player to Pursue his PhD in Math at MIT

John Urschel (NFL Player) to Pursue his PhD in Math at MIT (See link). This should be inspiring to all young learners in school/university who think it is not possible to mix sports and “seemingly” difficult subjects like Math that require lots of attention. Actually math is some kind of sport too, it requires practice to fully grasp the concepts. More interesting is what Math can do to help you get your sports right according to the University of Cambridge Math and Athletics Blog (See link).

How clear is this post?
By | January 28th, 2016|Uncategorized|0 Comments

Hello World

So this is my first post and any real “nerd” will know why I named my post “Hello World”. Many years ago, I dabbled in a little bit of computer programming and the first program you ever write, as kind of a rite of passage, is to make the screen say “Hello World”. Needless to say, it wasn’t always easy, but nothing worthwhile in life ever is. At times, I really did want to pull my hair out though, with things like syntax errors. Firstly, I had to work out what a syntax error was… Basically, in layman’s terms, it’s like making a grammar or punctuation error in an essay. Secondly, I needed to find this syntax error (or errors), most of the time it was a missing semi-colon, in my thirty or forty lines of code. Did I mention that the technology didn’t even give you a hint as to where you might begin to look for it.…

By | January 21st, 2016|Fun, Uncategorized|3 Comments

Mathematics: PISA could have said it all!

The Programme for International Students Assessment (PISA) has done various interesting research about mathematics and how it relates to society. You might find their findings interesting. Check them out: Click here

How clear is this post?
By | January 13th, 2016|Uncategorized|2 Comments

If Africa is to develop – Math has to be part of it!

Almost every nation in Africa is inspired by South Africa in terms of economic growth. But wait a minute, is South Africa really up there, well let’s see! In the recent world economic forum Global Information Technology Report of 2015, on mathematics and science education rankings, South Africa emerged as the worst country from the ranking. Refer to Link 0.

Actually, it is indeed astonishing that in the global rankings conducted by OECD every 2 years in the fields of science, reading and mathematics, there isn’t a single representative from Africa.  Is it because the continent is still under developed? well not directly but yes, it has something to do with it. The fact of the matter is that there seems to be a one-to-one correspondence between education (particularly math and science) and economic development as discussed in the OECD findings see: Link 1Link 2.

 

How clear is this post?
By | January 12th, 2016|Uncategorized|1 Comment

How much does a Dougie weigh on Jupiter?

I spent an evening with old friends just after Christmas. One of their sons is fascinated by Space, and always has great questions for me about it. This year he asked how much he would weigh on Jupiter. I admitted that I didn’t know the answer, but that a) It wasn’t a single number, as the surface of Jupiter (if one exists) is not very well understood and different heights in the atmosphere would give different weights and b) I would find out. It took a little Googling to find some information about the density profile of Jupiter, which includes information about Saturn too. The paper is the following, with abstract:

Taken from http://www.nature.com/nature/journal/v520/n7546/full/nature14278.html

Taken from http://www.nature.com/nature/journal/v520/n7546/full/nature14278.html

In this paper is the following pair of graphs:

Taken from http://www.nature.com/nature/journal/v520/n7546/images/nature14278-sf4.jpg

Which is all we need to calculate how much Dougie weighs on Jupiter. In the lower plot we see two different models of the density of Jupiter at different heights.…

By | January 10th, 2016|Fun, Uncategorized|0 Comments