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@ -3,107 +3,94 @@ category: Algorithms & Data Structures
name: Set theory name: Set theory
contributors: contributors:
--- ---
The set theory is a study for sets, their operations, and their properties. It is the basis of the whole mathematical system. Set theory is a branch of mathematics that studies sets, their operations, and their properties.
* A set is a collection of definite distinct items. * A set is a collection of disjoint items.
## Basic operators ## Basic symbols
These operators don't require a lot of text to describe.
* `∨` means or. ### Operators
* `∧` means and. * the union operator, `∪`, pronounced "cup", means "or";
* `,` separates the filters that determine the items in the set. * the intersection operator, `∩`, pronounced "cap", means "and";
* the exclusion operator, `\`, means "without";
* the compliment operator, `'`, means "the inverse of";
* the cross operator, `×`, means "the Cartesian product of".
## A brief history of the set theory ### Qualifiers
### Naive set theory * the colon qualifier, `:`, means "such that";
* Cantor invented the naive set theory. * the membership qualifier, `∈`, means "belongs to";
* It has lots of paradoxes and initiated the third mathematical crisis. * the subset qualifier, `⊆`, means "is a subset of";
* the proper subset qualifier, `⊂`, means "is a subset of but is not equal to".
### Axiomatic set theory ### Canonical sets
* It uses axioms to define the set theory. * `∅`, the empty set, i.e. the set containing no items;
* It prevents paradoxes from happening. * `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.
## Built-in sets There are a few caveats to mention regarding the canonical sets:
* `∅`, the set of no items. 1. Even though the empty set contains no items, the empty set is a subset of itself (and indeed every other set);
* `N`, the set of all natural numbers. `{0,1,2,3,…}` 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.
* `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.
``` ### Cardinality
A = {x|x∈N,x < 0}
A = ∅
∅ = {} (Sometimes)
|∅| = 0 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, `|...|`.
|{∅}| = 1
``` For example, if `S = { 1, 2, 4 }`, then `|S| = 3`.
### The Empty Set
* 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 1, i.e. `|∅| = 1`.
## Representing sets ## 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 ### Literal Sets
* Describes the features of all items in the set. Syntax: `{body|condtion}`
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
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 = { x : x is a vowel } = { a, e, i, o, u, y}
B = {x|x ∈ N, x < 10l} B = { x : x ∈ N, x < 10 } = { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 }
C = {x|x = 2k, k ∈ N} C = { x : x = 2k, k ∈ N } = { 0, 2, 4, 6, 8, ... }
C = {2x|x ∈ N}
``` ```
## Relations between sets Sometimes the predicate may "leak" into the subject, e.g.
### 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} D = { 2x : x ∈ N } = { 0, 2, 4, 6, 8, ... }
|A| = 3
B = {a,{b,c}}
|B| = 2
|∅| = 0 (it has no items)
``` ```
### Powerset ## Relations
* Let `A` be any set. The set that contains all possible subsets of `A` is called a powerset (written as `P(A)`).
### Membership
* 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`.
### Equality
* 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`.
## Special Sets
### The Power Set
* 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} P(A) = { x : x ⊆ A }
|A| = N, |P(A)| = 2^N
``` ```
## Set operations among two sets ## Set operations among two sets
@ -111,28 +98,28 @@ 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`. 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} A ∪ B = { x : x ∈ A ∪ x ∈ B }
``` ```
### Intersection ### 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`. 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} A ∩ B = { x : x ∈ A, x ∈ B }
``` ```
### Difference ### 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`. 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} A \ B = { x : x ∈ A, x ∉ B }
``` ```
### Symmetrical difference ### 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. 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 = { x : ((x ∈ A) ∩ (x ∉ B)) ∪ ((x ∈ B) ∩ (x ∉ A)) }
A △ B = (A \ B) ∪ (B \ A) A △ B = (A \ B) ∪ (B \ A)
``` ```
@ -141,22 +128,5 @@ 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`. 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 } 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
``` ```