 Studying computer science at the University of Cape Town. I intend to post about what I learn at MAM1000W the maths course I am doing at UCT. My posts may be about really simple topics in the beginning so bear with me, it gets interesting later on....

## Absolute values and inequalities

Things that I learnt today. Emphasis on the I, I couldn’t make for MAM1000W today.

• Absolute value definition
• Properties of absolute values
• Rules for inequalities

Absolute value

The absolute value of a number represents the distance between that number and $0$ on the real number line. Absolute value of a number $n$ is denoted by $|n|$ which is equal to $\sqrt{n^{2}}$ which is from the distance formula. Since it is the distance between $0$ and $n$ Hence $|n|=n$ if $n \geq 0$ and $|n|=-n$ if $n < 0$

Properties of Absolute values

1. $|nm| = |n||m|$
2. $|\frac{n}{m}| = \frac{|n|}{|m|}, (m \neq 0)$
3. $|n^{m}| = |n|^{m}$

Let $n > 0$ then

1. $|m| = n \iff m = \pm{n}$
2. $|m| < n \iff -n < m < n$
3. $|m| > n \iff m > n, m < -n$

Rules for inequalities

1. if $n then $n+p
2. if $n and $p then $n+p
3. if $n and $p>0$ then $np
4. if $n and $p<0$ then $np>mp$
5. if $0 then $\frac{1}{n}>\frac{1}{m}$
 How clear is this post?

## 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”|”.…

## 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.…