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disqus: abhyas29
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# Sets

###### tags: `Mathematics`
[All Mathematics Formula by Abhyas here](/@abhyas/maths_formula)

Please see [README](https://hackmd.io/@abhyas/maths_formula#README) if this is the first time you are here.

## Definition

Well defined collection of Distinct Elements

## Forms

### Roaster Form

Elements are listed in a comma sperated list bounded within curly braces

### Set Builder Form

Properties of the elements are defined in a particular form:

$A = \{x\, |\, x$ is a perfect square$\}$ read as "A is a set of all those $x$ such that $x$ is a perfect square."

- $A$ is the name of set.
- Within curly braces:
- $x$ is a varibale
- "$x$ is a perfect sqare" is the condition
- "$\,|\,$" (such that) seperated the variable from the condition

## Symbols

- Sets are denoted by upper case letters. eg. $A, P$ etc.
- If $x$ belongs to A, this can be denoted by $x \in A$.
- If $x$ doesnot belong to A, this can be denoted by $x \notin A$

## Notations

### Natural Numbers

$\mathbb{N} = \{1, 2, 3, 4, 5, 6 , ...\}$ upto infinity

$\mathbb{N}_{100}= \{1, 2, 3, 4, 5, ..., 98, 99, 100\}$ upto 100

NOTE: 0 is not a Natural Number

### Whole Numbers

$\mathbb{W}=\{0, 1, 2, 3, 4, ...\}$ upto inifinite

NOTE: 0 is part of Whole Numbers

### Integer

Set of all Integers: $\mathbb{Z} = \mathbb{I} = \{0, \pm1, \pm2, \pm3, \pm4, \pm5,...\}$

Set of all +ve Integers: $\mathbb{Z^+} = \{1, 2, 3, 4, 5, 6, 7,...\}$

Set of all -ve Integers: $\mathbb{Z^-} = \{-1, -2, -3, -4, -5, -6, -7,...\}$

Set of non-zero Integers: $\mathbb{Z}_0 = \{\pm1, \pm2, \pm3, \pm4, \pm5,...\}$

NOTE:
- Do not confuse with complex numebr $Z$.
- 0 is neither -ve nor +ve
- You can apply +ve, -ve and 0 on any other set. This was just easy to type. So no individual example for other sets. Plus doing so would make these notes too long.

### Rational Numbers

Set of rational numbers: $\mathbb{Q}=\{x|x= p/q$ and $p, q \in I$ and $q \ne 0\}$

Set of +ve, -ve and non zero: $\mathbb{Q}^+$, $\mathbb{Q}^-$ and $\mathbb{Q}_0$

### Irrational Numbers

Set of Irratonal Numbers = $\mathbb{R}-\mathbb{Q}$

### Real Numbers

All numbers except complex numbers

Set of all real numbers: $\mathbb{R}$

## Licensing and Links

[All Mathematics Formula by Abhays here](https://hackmd.io/@abhyas/maths_formula)

<a rel="license" href="http://creativecommons.org/licenses/by-nc/4.0/"><img alt="Creative Commons License" style="border-width:0" src="https://i.creativecommons.org/l/by-nc/4.0/88x31.png" /></a><br />This work is licensed under a <a rel="license" href="http://creativecommons.org/licenses/by-nc/4.0/">Creative Commons Attribution-NonCommercial 4.0 International License</a>.