# Sets

**Definition:** A set is an onordered collection of items with something in common

## Set vocabulary

**Element** - an item contained within a set

**Order** - The number of items in a set

**Subset **- A set where all the elements are also contained in another set

**Null set **- A set not containing any elements

**Universal set **- All possible values for the element, which are relevant.

**Disjoint sets **- No element between either set is shared.

## Set notation

Notation | Meaning |

$x\in A$ | The element x is contained in the set A |

$x\notin A$ | The element x is not contained in the set A |

$n\left(A\right)=x$ | The order of the set is x |

$A\subset B$ | Set A is a subset of set B |

$A\cap B$ | The intersection between set A and B. Contains all shared elements between both sets. |

$A\cup B$ | The union between set A and B. Contains all elements from both sets. |

$A\cup B=\varnothing $ | A and B are disjoint sets. |

$\varnothing $ | A null set or empty set |

$\xi $ or $U$ | The universe |

# Number Sets

There are different sytems in which we use to classify numbers and these are called the number sets.

Number set | Description | Examples | Symbol |

Real numbers | Any number which can be found on the number line. | $\mathrm{\pi},-1,5,\sqrt{2},\frac{7}{5}$ | $\mathrm{\mathbb{R}}$ |

Rational numbers | Any number which can be expressed by a fraction of two integers. | $-5,\frac{3}{5},8,\frac{2}{17}$ | $\mathrm{\mathbb{Q}}$ |

Irrational numbers | Any number which can't be expressed with a fraction. They have infinite non-repeating digts after the decimal place. | $\sqrt{2},\pi ,e,\sqrt{17}$ | $\mathrm{\mathbb{R}}/\mathrm{\mathbb{Q}}$ |

Integers | Any numbers which can be expressed without decimals or fractions. | $8,0,-100$ | $\mathrm{\mathbb{Z}}$ |

Whole numbers | Positive integers including 0 | $0,1,2,3$ | $\mathbb{W}$ |

Natural numbers | Positive integers excluding 0 | $1,2,3,4$ | $\mathrm{\mathbb{N}}$ |

Negative integers | All integers less than 0 | $-1,-2,-3$ | ${\mathrm{\mathbb{Z}}}^{-}$ |

## Diagram

# Exponents

An **Exponent** is the number that a number is raised to. This is also referred to as an indice.

## Rules of exponents

${a}^{m}\times {a}^{n}={a}^{m+n}$

${a}^{m}\xf7{a}^{n}={a}^{m-n}$

${\left({a}^{m}\right)}^{n}={a}^{mn}$

${a}^{-n}=\frac{1}{{a}^{n}}$

${a}^{\frac{1}{n}}=\sqrt[n]{a}$

${a}^{0}=1$

## Fractional exponents

A fractional exponent means that a number is being raised to the power of a fraction. It takes the form below:

**Form:** ${a}^{\frac{n}{m}}$

**Example: **${2}^{\frac{2}{3}}$

## Simplifying fractional exponents

We can easily simplify fractional indices by using the rules of indices.

${a}^{\frac{n}{m}}=({a}^{n}{)}^{\frac{1}{m}}=\sqrt[m]{{a}^{n}}$

# Logarithims

A** logarithm** is simply an algorithm which take an input and a base and outputs the exponent needed to raise the base by to produce the input.

## How to make a logarithm

A logarithm can be written using the form shown below:

${\mathrm{log}}_{n}\left(a\right)=b$

**n** - the base

**a **- the input

**b **- the output

For example: ${\mathrm{log}}_{2}\left(16\right)=4$

## Logarithm rules

Name | Rule |

Product rule | ${\mathrm{log}}_{b}\left(mn\right)={\mathrm{log}}_{b}\left(m\right)+{\mathrm{log}}_{b}\left(n\right)$ |

Quotient rule | ${\mathrm{log}}_{b}\left(\frac{m}{n}\right)={\mathrm{log}}_{b}\left(m\right)-{\mathrm{log}}_{b}\left(n\right)$ |

Power rule | ${\mathrm{log}}_{b}\left({m}^{n}\right)=n\xb7{\mathrm{log}}_{b}\left(m\right)$ |

Change of base rule | ${\mathrm{log}}_{a}\left(x\right)=\frac{{\mathrm{log}}_{b}\left(x\right)}{{\mathrm{log}}_{b}\left(a\right)}$ |

Equality rule | $if{\mathrm{log}}_{a}\left(x\right)={\mathrm{log}}_{a}\left(y\right),thenx=y$ |

# Editors

- joeClinton - 387 words.

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