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diff --git a/set-theory.html.markdown b/set-theory.html.markdown index c6bc39c5..6fb657ed 100644 --- a/set-theory.html.markdown +++ b/set-theory.html.markdown @@ -3,107 +3,94 @@ 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. +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 -These operators don't require a lot of text to describe. +## Basic symbols -* `∨` means or. -* `∧` means and. -* `,` separates the filters that determine the items in the set. +### Operators +* the union operator, `∪`, pronounced "cup", means "or"; +* 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 -### Naive set theory -* Cantor invented the naive set theory. -* It has lots of paradoxes and initiated the third mathematical crisis. +### Qualifiers +* the colon qualifier, `:`, means "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". -### Axiomatic set theory -* It uses axioms to define the set theory. -* It prevents paradoxes from happening. +### Canonical sets +* `∅`, 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. -## 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. +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 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) +### Cardinality -|∅| = 0 -|{∅}| = 1 -``` +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 +* 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 -### 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}` +### Literal Sets + +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} -B = {x|x ∈ N, x < 10l} -C = {x|x = 2k, k ∈ N} -C = {2x|x ∈ N} +A = { x : x is a vowel } = { a, e, i, o, u, y} +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, ... } ``` -## 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` +Sometimes the predicate may "leak" into the subject, e.g. -### 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` +``` +D = { 2x : x ∈ N } = { 0, 2, 4, 6, 8, ... } +``` -### 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`). +## Relations -### 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`. +### Membership -### 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`). +* 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`. -## 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. +### Equality -``` -A = {A,B,C} -|A| = 3 -B = {a,{b,c}} -|B| = 2 -|∅| = 0 (it has no items) -``` +* 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`. -### Powerset -* Let `A` be any set. The set that contains all possible subsets of `A` is called a powerset (written as `P(A)`). +## Special Sets -``` -P(A) = {x|x ⊆ A} +### 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. -|A| = N, |P(A)| = 2^N +``` +P(A) = { x : x ⊆ A } ``` ## 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`. ``` -A ∪ B = {x|x∈A∨x∈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} +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} +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 = { x : ((x ∈ A) ∩ (x ∉ B)) ∪ ((x ∈ B) ∩ (x ∉ 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`. ``` -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 +A × B = { (x, y) | x ∈ A, y ∈ B } ``` |