The set theory is a study for sets, their operations, and their properties. It is the basis of the whole mathematical system.

- A set is a collection of definite distinct items.

These operators don’t require a lot of text to describe.

`∨`

means or.`∧`

means and.`,`

separates the filters that determine the items in the set.

- Cantor invented the naive set theory.
- It has lots of paradoxes and initiated the third mathematical crisis.

- It uses axioms to define the set theory.
- It prevents paradoxes from happening.

`∅`

, the set of no items.`N`

, the set of all natural numbers.`{0,1,2,3,…}`

`Z`

, the set of all integers.`{…,-2,-1,0,1,2,…}`

`Q`

, the set of all rational numbers.`R`

, the set of all real numbers.

- The set containing no items is called the empty set. Representation:
`∅`

- The empty set can be described as
`∅ = {x|x ≠ x}`

- The empty set is always unique.
- The empty set is the subset of all sets.

```
A = {x|x∈N,x < 0}
A = ∅
∅ = {} (Sometimes)
|∅| = 0
|{∅}| = 1
```

- List all items of the set, e.g.
`A = {a,b,c,d}`

- List some of the items of the set. Ignored items are represented with
`…`

. E.g.`B = {2,4,6,8,10,…}`

- Describes the features of all items in the set. Syntax:
`{body|condtion}`

```
A = {x|x is a vowel}
B = {x|x ∈ N, x < 10l}
C = {x|x = 2k, k ∈ N}
C = {2x|x ∈ N}
```

- If the value
`a`

is one of the items of the set`A`

,`a`

belongs to`A`

. Representation:`a∈A`

- If the value
`a`

is not one of the items of the set`A`

,`a`

does not belong to`A`

. Representation:`a∉A`

- If all items in a set are exactly the same to another set, they are equal. Representation:
`a=b`

- Items in a set are not order sensitive.
`{1,2,3,4}={2,3,1,4}`

- Items in a set are unique.
`{1,2,2,3,4,3,4,2}={1,2,3,4}`

- Two sets are equal if and only if all of their items are exactly equal to each other. Representation:
`A=B`

. Otherwise, they are not equal. Representation:`A≠B`

. `A=B`

if`A ⊆ B`

and`B ⊆ A`

- If the set A contains an item
`x`

,`x`

belongs to A (`x∈A`

). - Otherwise,
`x`

does not belong to A (`x∉A`

).

- If all items in a set
`B`

are items of set`A`

, we say that`B`

is a subset of`A`

(`B⊆A`

). - If B is not a subset of A, the representation is
`B⊈A`

.

- If
`B ⊆ A`

and`B ≠ A`

, B is a proper subset of A (`B ⊂ A`

). Otherwise, B is not a proper subset of A (`B ⊄ A`

).

- The number of items in a set is called the base number of that set. Representation:
`|A|`

- If the base number of the set is finite, this set is a finite set.
- If the base number of the set is infinite, this set is an infinite set.

```
A = {A,B,C}
|A| = 3
B = {a,{b,c}}
|B| = 2
|∅| = 0 (it has no items)
```

- Let
`A`

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

is called a powerset (written as`P(A)`

).

```
P(A) = {x|x ⊆ A}
|A| = N, |P(A)| = 2^N
```

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 }
```

Better known as “flattening” of a set of sets.

```
∪A = {x|X∈A,x∈X}
∪A={a,b,c,d,e,f}
∪B={a}
∪C=a∪{c,d}
```

```
∩ A = A1 ∩ A2 ∩ … ∩ An
```

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Originally contributed by , and updated by 1 contributor(s).