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+---
+category: Algorithms & Data Structures
+name: Set theory
+contributors:
+---
+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.
+
+## Basic operators
+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.
+
+## A brief history of the set theory
+### Naive set theory
+* Cantor invented the naive set theory.
+* It has lots of paradoxes and initiated the third mathematical crisis.
+
+### Axiomatic set theory
+* It uses axioms to define the set theory.
+* It prevents paradoxes from happening.
+
+## Built-in sets
+* `∅`, 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 empty set
+* 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
+```
+## Representing sets
+### Enumeration
+* 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,…}`
+
+### Description
+* 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}
+```
+
+## Relations between sets
+### Belongs to
+* 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`
+
+### Equals
+* 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`
+
+### Belongs to
+* If the set A contains an item `x`, `x` belongs to A (`x∈A`).
+* Otherwise, `x` does not belong to A (`x∉A`).
+
+### Subsets
+* 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`.
+
+### Proper subsets
+* 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`).
+
+## Set operations
+### Base number
+* 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)
+```
+
+### Powerset
+* 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
+```
+
+## Set operations among two sets
+### Union
+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}
+```
+### Intersection
+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}
+```
+### Difference
+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}
+```
+### Symmetrical difference
+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)
+```
+### Cartesian product
+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 }
+```
+## "Generalized" operations
+### General union
+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}
+```
+### General intersection
+```
+∩ A = A1 ∩ A2 ∩ … ∩ An
+```