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144 lines
4.5 KiB
Markdown
144 lines
4.5 KiB
Markdown
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---
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category: Algorithms & Data Structures
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name: Set theory
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contributors:
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---
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The set theory is a study for sets, their operations, and their properties. It is the basis of the whole mathematical system.
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* A set is a collection of definite distinct items.
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## Basic operators
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These operators don't require a lot of text to describe.
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* `∨` means or.
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* `∧` means and.
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* `,` separates the filters that determine the items in the set.
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## A brief history of the set theory
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### Naive set theory
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* Cantor invented the naive set theory.
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* It has lots of paradoxes and initiated the third mathematical crisis.
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### Axiomatic set theory
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* It uses axioms to define the set theory.
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* It prevents paradoxes from happening.
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## Built-in sets
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* `∅`, the set of no items.
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* `N`, the set of all natural numbers. `{0,1,2,3,…}`
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* `Z`, the set of all integers. `{…,-2,-1,0,1,2,…}`
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* `Q`, the set of all rational numbers.
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* `R`, the set of all real numbers.
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### The empty set
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* The set containing no items is called the empty set. Representation: `∅`
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* The empty set can be described as `∅ = {x|x ≠ x}`
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* The empty set is always unique.
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* The empty set is the subset of all sets.
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```
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A = {x|x∈N,x < 0}
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A = ∅
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∅ = {} (Sometimes)
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|∅| = 0
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|{∅}| = 1
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```
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## Representing sets
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### Enumeration
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* List all items of the set, e.g. `A = {a,b,c,d}`
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* List some of the items of the set. Ignored items are represented with `…`. E.g. `B = {2,4,6,8,10,…}`
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### Description
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* Describes the features of all items in the set. Syntax: `{body|condtion}`
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```
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A = {x|x is a vowel}
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B = {x|x ∈ N, x < 10l}
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C = {x|x = 2k, k ∈ N}
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C = {2x|x ∈ N}
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```
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## Relations between sets
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### Belongs to
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* If the value `a` is one of the items of the set `A`, `a` belongs to `A`. Representation: `a∈A`
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* If the value `a` is not one of the items of the set `A`, `a` does not belong to `A`. Representation: `a∉A`
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### Equals
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* If all items in a set are exactly the same to another set, they are equal. Representation: `a=b`
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* Items in a set are not order sensitive. `{1,2,3,4}={2,3,1,4}`
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* Items in a set are unique. `{1,2,2,3,4,3,4,2}={1,2,3,4}`
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* 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`.
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* `A=B` if `A ⊆ B` and `B ⊆ A`
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### Belongs to
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* If the set A contains an item `x`, `x` belongs to A (`x∈A`).
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* Otherwise, `x` does not belong to A (`x∉A`).
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### Subsets
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* If all items in a set `B` are items of set `A`, we say that `B` is a subset of `A` (`B⊆A`).
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* If B is not a subset of A, the representation is `B⊈A`.
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### Proper subsets
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* 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`).
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## Set operations
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### Base number
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* The number of items in a set is called the base number of that set. Representation: `|A|`
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* If the base number of the set is finite, this set is a finite set.
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* If the base number of the set is infinite, this set is an infinite set.
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```
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A = {A,B,C}
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|A| = 3
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B = {a,{b,c}}
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|B| = 2
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|∅| = 0 (it has no items)
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```
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### Powerset
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* Let `A` be any set. The set that contains all possible subsets of `A` is called a powerset (written as `P(A)`).
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```
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P(A) = {x|x ⊆ A}
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|A| = N, |P(A)| = 2^N
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```
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## Set operations among two sets
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### Union
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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`.
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```
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A ∪ B = {x|x∈A∨x∈B}
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```
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### Intersection
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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`.
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```
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A ∩ B = {x|x∈A,x∈B}
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```
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### Difference
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Given two sets `A` and `B`, the set difference of `A` with `B` is every item in `A` that does not belong to `B`.
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```
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A \ B = {x|x∈A,x∉B}
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```
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### Symmetrical difference
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Given two sets `A` and `B`, the symmetrical difference is all items among `A` and `B` that doesn't appear in their intersections.
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```
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A △ B = {x|(x∈A∧x∉B)∨(x∈B∧x∉A)}
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A △ B = (A \ B) ∪ (B \ A)
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```
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### Cartesian product
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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`.
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```
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A × B = { {x, y} | x ∈ A, y ∈ B }
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```
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## "Generalized" operations
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### General union
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Better known as "flattening" of a set of sets.
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```
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∪A = {x|X∈A,x∈X}
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∪A={a,b,c,d,e,f}
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∪B={a}
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∪C=a∪{c,d}
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```
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### General intersection
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```
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∩ A = A1 ∩ A2 ∩ … ∩ An
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```
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