# Sets

A set is one of the most fundamental concepts in mathematics. Sets can be taught at an elementary level all the way through higher level mathematics.

A **set** is defined as a group or collection of distinct objects. The elements
of a set can be anything: numbers, people, letters, etc. The way we usual denote
sets is by giving them capital letters for a name.

Given set A and B

A is equal to B *if and only if* they have precisely the same elements.

## Notation

There are different ways of notating and describing sets and more specifically,
members of a set. One way is **intensional definition**. We can literally describe
what the set contains.

Another way is by **extension**, which means listing each member of the set and
enclosing them with brackets. This is also called *strict enumeration*.

Each element of a set must be unique, and the order of the elements in the set does not matter.

When we have sets with many elements, we can abbreviate the list to denote an extension
of numbers in the same pattern as the numbers before it. This is sometimes called
*pattern enumeration*. If we were to list all of the odd numbers up to 100,
we would have

If it was an infinite list, we can use it at the end. Here is the list of all positive odd numbers

Set's can also be defined by a rule. The set A of the positive integers that are perfect squares less than 10 can be notated as

Set names are subjective, and different definitions can be the same set of numbers. It is up to the mathematician to accurately describe the set.

## Subsets

Subsets introduce the notion of membership of sets - when one set is contained in another. If every member of A is also a member of set B, then A is a subset of B, which is written as

The relationship is called *inclusion* or *containment*. The set **{1,2,4}**
would be a subset of **{1,2,3,4}**. The empty set is also a subset of every set
and every set is a subset of itself.

When **A is a subset of B** and **B is a subset of A**, then **A = B**.

Let's do an example illustrating sets using number classifications.

We have all of the following types of numbers:

Every type of a number classification is a set which contains elements that have unique traits, and are also subsets of other classifications. We can deduce from this classification tree that

## Basic Operations

### Unions

A Union is an operation that adds two sets together. The union of **A** and **B**
is denoted as

which is the set of all elements which are members of either A *or* B.

Like the Real Numbers, sets have operational properties.

### Intersection

The intersection of two sets is the elements that both sets have in common. The
intersection of **A** and **B** is denoted as

The elements must be contained in both A *and* B.

Intersections have similar properties to Unions.

### Complements

A complement is the difference of sets. The relative complement of B in A is denoted by

which is the set of all elements which are members of A but not members of B.

Complements have a different notation when considered within a *universal set U*,
where the complement of A is every element in U except A.

If **U** is the set of natural numbers and **A = {1,2,3}**, then **A'**
would be **{4,5,6...}**.

Here are some basic properties of complements

### Cartesian Product

New sets can be constructed by multiplying sets together. The Cartesian Product is similar to matrix multiplication, where each element of a set gets paired with every element of the other set. The Cartesian product of A and B is denoted by

Here are a few properties of the Cartesian product