Set theory is a branch of mathematics that studies sets, their operations, and their properties.

- A set is a collection of disjoint items.

- the union operator,
`∪`

, pronounced “cup”, means “or”; - the intersection operator,
`∩`

, pronounced “cap”, means “and”; - the exclusion operator,
`\`

, means “without”; - the complement operator,
`'`

, means “the inverse of”; - the cross operator,
`×`

, means “the Cartesian product of”.

- the colon,
`:`

, or the vertical bar`|`

qualifiers are interchangeable and mean “such that”; - the membership qualifier,
`∈`

, means “belongs to”; - the subset qualifier,
`⊆`

, means “is a subset of”; - the proper subset qualifier,
`⊂`

, means “is a subset of but is not equal to”.

`∅`

, the empty set, i.e. the set containing no items;`ℕ`

, the set of all natural numbers;`ℤ`

, the set of all integers;`ℚ`

, the set of all rational numbers;`ℝ`

, the set of all real numbers.

There are a few caveats to mention regarding the canonical sets: 1. Even though the empty set contains no items, the empty set is a subset of itself (and indeed every other set); 2. Mathematicians generally do not universally agree on whether zero is a natural number, and textbooks will typically explicitly state whether or not the author considers zero to be a natural number.

The cardinality, or size, of a set is determined by the number of items in the set. The cardinality operator is given by a double pipe, `|...|`

.

For example, if `S = { 1, 2, 4 }`

, then `|S| = 3`

.

- The empty set can be constructed in set builder notation using impossible conditions, e.g.
`∅ = { x : x ≠ x }`

, or`∅ = { x : x ∈ N, x < 0 }`

; - the empty set is always unique (i.e. there is one and only one empty set);
- the empty set is a subset of all sets;
- the cardinality of the empty set is 0, i.e.
`|∅| = 0`

.

A set can be constructed literally by supplying a complete list of objects contained in the set. For example, `S = { a, b, c, d }`

.

Long lists may be shortened with ellipses as long as the context is clear. For example, `E = { 2, 4, 6, 8, ... }`

is clearly the set of all even numbers, containing an infinite number of objects, even though we’ve only explicitly written four of them.

Set builder notation is a more descriptive way of constructing a set. It relies on a *subject* and a *predicate* such that `S = { subject : predicate }`

. For example,

```
A = { x : x is a vowel } = { a, e, i, o, u }
B = { x : x ∈ N, x < 10 } = { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 }
C = { x : x = 2k, k ∈ N } = { 0, 2, 4, 6, 8, ... }
```

Sometimes the predicate may “leak” into the subject, e.g.

```
D = { 2x : x ∈ N } = { 0, 2, 4, 6, 8, ... }
```

- If the value
`a`

is contained in the set`A`

, then we say`a`

belongs to`A`

and represent this symbolically as`a ∈ A`

. - If the value
`a`

is not contained in the set`A`

, then we say`a`

does not belong to`A`

and represent this symbolically as`a ∉ A`

.

- If two sets contain the same items then we say the sets are equal, e.g.
`A = B`

. - Order does not matter when determining set equality, e.g.
`{ 1, 2, 3, 4 } = { 2, 3, 1, 4 }`

. - Sets are disjoint, meaning elements cannot be repeated, e.g.
`{ 1, 2, 2, 3, 4, 3, 4, 2 } = { 1, 2, 3, 4 }`

. - Two sets
`A`

and`B`

are equal if and only if`A ⊆ B`

and`B ⊆ A`

.

- Let
`A`

be any set. The set that contains all possible subsets of`A`

is called a “power set” and is written as`P(A)`

. If the set`A`

contains`n`

elements, then`P(A)`

contains`2^n`

elements.

```
P(A) = { x : x ⊆ A }
```

Given two sets `A`

and `B`

, the union of the two sets are the items that appear in either `A`

or `B`

, written as `A ∪ B`

.

```
A ∪ B = { x : x ∈ A ∪ x ∈ B }
```

Given two sets `A`

and `B`

, the intersection of the two sets are the items that appear in both `A`

and `B`

, written as `A ∩ B`

.

```
A ∩ B = { x : x ∈ A, x ∈ B }
```

Given two sets `A`

and `B`

, the set difference of `A`

with `B`

is every item in `A`

that does not belong to `B`

.

```
A \ B = { x : x ∈ A, x ∉ B }
```

Given two sets `A`

and `B`

, the symmetrical difference is all items among `A`

and `B`

that doesn’t appear in their intersections.

```
A △ B = { x : ((x ∈ A) ∩ (x ∉ B)) ∪ ((x ∈ B) ∩ (x ∉ A)) }
A △ B = (A \ B) ∪ (B \ A)
```

Given two sets `A`

and `B`

, the cartesian product between `A`

and `B`

consists of a set containing all combinations of items of `A`

and `B`

.

```
A × B = { (x, y) | x ∈ A, y ∈ B }
```

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Originally contributed by Andrew Ryan Davis, and updated by 9 contributor(s).