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”|”. But it still carries the same meaning.

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

moreover

$\mathit{D}=\{e : \mathit{C} (e) \}$

means $e$ satisfies the characteristic $\mathit{C} (e)$

Different kinds of sets:

1.The empty set

The set containing no elements at all is denoted: $\emptyset$

2.Integers

Every number without a decimal or fractional part, denoted by $\mathbb{Z}$

3.Natural numbers

Represents all nonnegative integers: $\mathbb{N}$

4.Counting numbers

All positive integers: $\mathbb{N}^{+}$

5.Rational numbers

Every integer that can be written in the form: $\frac{a}{b}$ where $b \neq 0$ denoted by $\mathbb{Q}$.

6.Real numbers

All the numbers on the number line, denoted by $\mathbb{R}$

7.Irrational numbers

Real numbers that cannot be written in the form $\frac{a}{b}$ where $b \neq 0$ denoted by $\mathbb{Q} ^ {\prime}$

Intervals

let $p$ and $q$ belong to $\mathbb{R}$, then

$(p,q) = \{ y \in \mathbb{R} : p < y < q\}$

$[p,q) = \{ y \in \mathbb{R} : p \leq y < q\}$

$(p,q] = \{ y \in \mathbb{R} : p < y \leq q\}$

$[p,q] = \{ y \in \mathbb{R} : p \leq y \leq q\}$

$(p,\infty) = \{ y \in \mathbb{R} : y > p\}$

$(- \infty,p) = \{ y \in \mathbb{R} : y < p \}$

$[p, \infty) = \{ y \in \mathbb{R} : y \geq p\}$

$(- \infty,p] = \{ y \in \mathbb{R} : y \leq p\}$

$(- \infty, \infty) = \mathbb{R}$

Combining sets

$\mathit{A} \cup \mathit{B}$

is all the elements in $\mathit{A}$ or $\mathit{B}$

$\mathit{A} \cap \mathit{B}$

is all the elements in $\mathit{A}$ and $\mathit{B}$

if

$\mathit{A} \cap \mathit{B} = \emptyset$

then $\mathit{A}$ and $\mathit{B}$ are called a disjoint.

Set notation in functions

$\mathit{F} : \mathit{D} \to \mathbb{R}$

means that every element in the set $\mathit{D}$ is assigned to an element in the set $\mathbb{R}$ by the function $\mathit{F}$.

My first time using LaTeX. And I survived the fires of UCT.

 How clear is this post?