# Sets

## Set Definition:

A collection of objects.
Examples include letters, number, fruits.

## Set Notation

{1, 2, 3, 4, 5}
Use braces {} to enclose a set

Use 3 dots for infinity
{1, 2, 3, ...}

Use capital letters for sets
S = {1, 2, 3}

## Examples of Sets:

Letters of the alphabet {a, b, c}
Coins: {penny, nickel, dime}
Counting Numbers {1, 2, 3, ...}
Positive Even Numbers {2, 4, 6, ...}

## Elements of sets:

Each element separated by commas
Elements use the element symbol ∈
In the set S = {1, 2, 3}
1 ∈ S, 2 ∈ S, 3 ∈ S

## Cardinality of a set:

Measures how many elements
Given a set S, cardinality = |S|
With S = {1, 2, 3}, |S| = 3
since S has 3 elements

## Special Type of Sets:

The Universal Set U has every element
The empty set ∅ has no elements

## Complement of a set:

Everything not in the set but in U
Write this as S' or SC
Given U = {1, 2, 3, 4, 5} and S = {1, 2, 3}
SC = {4, 5} since they are in U not in S

## Finite sets:

Finite sets have countable elements.
{1, 2, 3} for example has 3 elements

## Infinite sets:

Infinite sets have uncountable elements
They go on forever using ...symbol
{1, 2, 3, ...}

## Set Notes:

Element order does not matter
{1, 2, 3} = {3, 2, 1}

### How does the Sets Calculator work?

Free Sets Calculator - This lesson walks you through what a set is, how to write a set, elements of a set, types of sets, cardinality of a set, complement of a set.

### What 5 formulas are used for the Sets Calculator?

Set is denoted with braces and commas separating elements {1, 2, 3}
Elements are members of items in a set S = {1, 2, 3}, 1 ∈ S
S = {1, 2, 3}, the cardinality |S| = 3 since S has 3 elements
The empty set ∅ contains no elements
Given a Universal Set U = {1, 2, 3, 4, 5} and S = {1, 2, 3} SC = {4, 5} since they are everything in the Universal set not in S

For more math formulas, check out our Formula Dossier

### What 11 concepts are covered in the Sets Calculator?

cardinality
a measure of the number of elements of the set
coefficient
a numerical or constant quantity placed before and multiplying the variable in an algebraic expression
difference
the result of one of the important mathematical operations, which is obtained by subtracting two numbers
element
an element (or member) of a set is any one of the distinct objects that belong to that set. In chemistry, any substance that cannot be decomposed into simpler substances by ordinary chemical processes.
index
an indicator, sign, or measure of something
intersection
the set containing all elements of A that also belong to B or equivalently, all elements of B that also belong to A.
A ∩ B
product
The answer when two or more values are multiplied together
set
a collection of different things; a set contains elements or members, which can be mathematical objects of any kind
sets
subset
A is a subset of B if all elements of the set A are elements of the set B
union
Combine the elements of two or more sets