Set (mathematics)

In mathematics, a set is a collection of distinct objects, considered as an object in its own right. For example, the numbers 2, 4, and 6 are distinct objects when considered separately, but when they are considered collectively they form a single set of size three, written {2, 4, 6}. The concept of a set is one of the most fundamental in mathematics. Developed at the end of the 19th century, set theory is now a ubiquitous part of mathematics, and can be used as a foundation from which nearly all of mathematics can be derived. In mathematics education, elementary topics from set theory such as Venn diagrams are taught at a young age, while more advanced concepts are taught as part of a university degree.

The German word Menge, rendered as "set" in English, was coined by Bernard Bolzano in his work The Paradoxes of the Infinite.

Example of a set
A set of polygons in an Euler diagram

Definition

Passage with the set definition of Georg Cantor
Passage with a translation of the original set definition of Georg Cantor. The German word Menge for set is translated with aggregate here.

A set is a well-defined collection of distinct objects. The objects that make up a set (also known as the set's elements or members) can be anything: numbers, people, letters of the alphabet, other sets, and so on. Georg Cantor, one of the founders of set theory, gave the following definition of a set at the beginning of his Beiträge zur Begründung der transfiniten Mengenlehre:[1]

A set is a gathering together into a whole of definite, distinct objects of our perception [Anschauung] or of our thought—which are called elements of the set.

Sets are conventionally denoted with capital letters. Sets A and B are equal if and only if they have precisely the same elements.[2]

For technical reasons, Cantor's definition turned out to be inadequate; today, in contexts where more rigor is required, one can use axiomatic set theory, in which the notion of a "set" is taken as a primitive notion and the properties of sets are defined by a collection of axioms. The most basic properties are that a set can have elements, and that two sets are equal (one and the same) if and only if every element of each set is an element of the other; this property is called the extensionality of sets.

Describing sets

There are two ways of describing, or specifying the members of, a set. One way is by intensional definition, using a rule or semantic description:

A is the set whose members are the first four positive integers.
B is the set of colors of the French flag.

The second way is by extension – that is, listing each member of the set. An extensional definition is denoted by enclosing the list of members in curly brackets:

C = {4, 2, 1, 3}
D = {blue, white, red}.

One often has the choice of specifying a set either intensionally or extensionally. In the examples above, for instance, A = C and B = D.

In an extensional definition, a set member can be listed two or more times, for example, {11, 6, 6}. However, per extensionality, two definitions of sets which differ only in that one of the definitions lists members multiple times define the same set. Hence, the set {11, 6, 6} is identical to the set {11, 6}. Moreover, the order in which the elements of a set are listed is irrelevant (unlike for a sequence or tuple), so {6, 11} is yet again the same set.

For sets with many elements, the enumeration of members can be abbreviated. For instance, the set of the first thousand positive integers may be specified extensionally as

{1, 2, 3, ..., 1000},

where the ellipsis ("...") indicates that the list continues in the obvious way.

The notation with braces may also be used in an intensional specification of a set. In this usage, the braces have the meaning "the set of all ...". So, E = {playing card suits} is the set whose four members are spades, diamonds, hearts, and clubs. A more general form of this is set-builder notation, through which, for instance, the set F of the twenty smallest integers that are four less than a perfect square can be denoted

In this notation, the colon (":") means "such that", and the description can be interpreted as "F is the set of all numbers of the form n2 − 4, such that n is a whole number in the range from 0 to 19 inclusive." Sometimes the vertical bar ("|") is used instead of the colon.

Membership

If B is a set and x is one of the objects of B, this is denoted as xB, and is read as "x is an element of B", as "x belongs to B", or short "x is in B". If y is not a member of B then this is written as yB, read as "y is not an element of B", or as analogous negated forms.

For example, with respect to the sets A = {1, 2, 3, 4}, B = {blue, white, red}, and F = {n2 − 4 : n is an integer, and 0 ≤ n ≤ 19} defined above,

4 ∈ A and 12 ∈ F; and
9 ∉ F and green ∉ B.

Subsets

If every element of set A is also in B, then A is said to be a subset of B, written AB (pronounced A is contained in B). Equivalently, one can write BA, read as B is a superset of A, B includes A, or B contains A. The relationship between sets established by ⊆ is called inclusion or containment, and is given also for equal sets, that is, equality of sets is the same as mutual containment in each other: AB and BA is equivalent to A = B.

If A is a subset of, but not equal to, B, then A is called a proper subset of B, written AB, or simply AB (A is a proper subset of B), or BA (B is a proper superset of A, BA).

The expressions AB and BA are used differently by different authors; some authors use them to mean the same as AB (respectively BA), whereas others use them to mean the same as AB (respectively BA).

Venn A subset B
A is a subset of B

Examples:

  • The set of all men is a proper subset of the set of all people.
  • {1, 3} ⊆ {1, 2, 3, 4}.
  • {1, 2, 3, 4} ⊆ {1, 2, 3, 4}.

The empty set is a subset of every set, and every set is a subset of itself:

  • ∅ ⊆ A.
  • AA.

The above characterization of set equality can be used to show that two sets described differently are, in fact, equal:

  • A = B if and only if AB and BA.

A partition of a set S is a set of nonempty subsets of S such that every element x in S is in exactly one of these subsets, that is, any two sets of the partition contain no element in common, they are said to be disjoint, and the union of all elements of the partition that are sets themselves, make up S.

Power sets

The power set of a set S is the set of all subsets of S. The power set contains S itself and the empty set because these are both subsets of S. For example, the power set of the set {1, 2, 3} is {{1, 2, 3}, {1, 2}, {1, 3}, {2, 3}, {1}, {2}, {3}, ∅}. The power set of a set S is usually written as P(S).

The power set of a finite set with n elements has 2n elements. For example, the set {1, 2, 3} contains three elements, and the power set shown above contains 23 = 8 elements.

The power set of an infinite (either countable or uncountable) set is always uncountable. Moreover, the power set of a set is always strictly "bigger" than the original set in the sense that there is no way to pair every element of S with exactly one element of P(S). (There is never an onto map or surjection from S onto P(S).)

Every partition of a set S is a subset of the powerset of S.

Cardinality

The cardinality of a set S, denoted |S|, is the number of members of S. For example, if B = {blue, white, red}, then |B| = 3.

There is a unique set with no members, called the empty set (or the null set), which is denoted by the symbol ∅ (other notations are used; see empty set). The cardinality of the empty set is zero. For example, the set of all three-sided squares has zero members and thus is the empty set. Though it may seem trivial, the empty set, like the number zero, is important in mathematics. Indeed, the existence of this set is one of the fundamental concepts of axiomatic set theory.

Some sets have infinite cardinality. The set N of natural numbers, for instance, is infinite. Some infinite cardinalities are greater than others. For instance, the set of real numbers has greater cardinality than the set of natural numbers. However, it can be shown that the cardinality of (which is to say, the number of points on) a straight line is the same as the cardinality of any segment of that line, of the entire plane, and indeed of any finite-dimensional Euclidean space.

Special sets

There are some sets or kinds of sets that hold great mathematical importance and are referred to with such regularity that they have acquired special names and notational conventions to identify them. One of these is the empty set, denoted { } or ∅. A set with exactly one element, x, is a unit set, or singleton, {x}.[2]

Many of these sets are represented using blackboard bold or bold typeface. Special sets of numbers include

  • P or ℙ, denoting the set of all primes: P = {2, 3, 5, 7, 11, 13, 17, ...}.
  • N or , denoting the set of all natural numbers: N = {0, 1, 2, 3, ...} (sometimes defined excluding 0).
  • Z or , denoting the set of all integers (whether positive, negative or zero): Z = {..., −2, −1, 0, 1, 2, ...}.
  • Q or ℚ, denoting the set of all rational numbers (that is, the set of all proper and improper fractions): Q = {a/b | a, bZ, b ≠ 0}. For example, 1/4 ∈ Q and 11/6 ∈ Q. All integers are in this set since every integer a can be expressed as the fraction a/1 (ZQ).
  • R or , denoting the set of all real numbers. This set includes all rational numbers, together with all irrational numbers (that is, algebraic numbers that cannot be rewritten as fractions such as 2, as well as transcendental numbers such as π, e).
  • C or ℂ, denoting the set of all complex numbers: C = {a + bi|a, bR}. For example, 1 + 2iC.
  • H or ℍ, denoting the set of all quaternions: H = {a + bi + cj + dk|a, b, c, dR}. For example, 1 + i + 2jkH.

Each of the above sets of numbers has an infinite number of elements, and each can be considered to be a proper subset of the sets listed below it. The primes are used less frequently than the others outside of number theory and related fields.

Positive and negative sets are sometimes denoted by superscript plus and minus signs, respectively. For example, ℚ+ represents the set of positive rational numbers.

Basic operations

There are several fundamental operations for constructing new sets from given sets.

Unions

Venn0111
The union of A and B, denoted AB

Two sets can be "added" together. The union of A and B, denoted by A ∪ B, is the set of all things that are members of either A or B.

Examples:

  • {1, 2} ∪ {1, 2} = {1, 2}.
  • {1, 2} ∪ {2, 3} = {1, 2, 3}.
  • {1, 2, 3} ∪ {3, 4, 5} = {1, 2, 3, 4, 5}

Some basic properties of unions:

  • AB = BA.
  • A ∪ (BC) = (AB) ∪ C.
  • A ⊆ (AB).
  • AA = A.
  • AU = U.
  • A ∪ ∅ = A.
  • AB if and only if AB = B.

Intersections

A new set can also be constructed by determining which members two sets have "in common". The intersection of A and B, denoted by AB, is the set of all things that are members of both A and B. If AB = ∅, then A and B are said to be disjoint.

Venn0001
The intersection of A and B, denoted AB.

Examples:

  • {1, 2} ∩ {1, 2} = {1, 2}.
  • {1, 2} ∩ {2, 3} = {2}.

Some basic properties of intersections:

  • AB = BA.
  • A ∩ (BC) = (AB) ∩ C.
  • ABA.
  • AA = A.
  • AU = A.
  • A ∩ ∅ = ∅.
  • AB if and only if AB = A.

Complements

Venn0100
The relative complement
of B in A
Venn1010
The complement of A in U
Venn0110
The symmetric difference of A and B

Two sets can also be "subtracted". The relative complement of B in A (also called the set-theoretic difference of A and B), denoted by A \ B (or AB), is the set of all elements that are members of A but not members of B. Note that it is valid to "subtract" members of a set that are not in the set, such as removing the element green from the set {1, 2, 3}; doing so has no effect.

In certain settings all sets under discussion are considered to be subsets of a given universal set U. In such cases, U \ A is called the absolute complement or simply complement of A, and is denoted by A′.

  • A′ = U \ A

Examples:

  • {1, 2} \ {1, 2} = ∅.
  • {1, 2, 3, 4} \ {1, 3} = {2, 4}.
  • If U is the set of integers, E is the set of even integers, and O is the set of odd integers, then U \ E = E′ = O.

Some basic properties of complements:

  • A \ BB \ A for AB.
  • AA′ = U.
  • AA′ = ∅.
  • (A′)′ = A.
  • ∅ \ A = ∅.
  • A \ ∅ = A.
  • A \ A = ∅.
  • A \ U = ∅.
  • A \ A′ = A and A′ \ A = A′.
  • U′ = ∅ and ∅′ = U.
  • A \ B = AB.
  • if AB then A \ B = ∅.

An extension of the complement is the symmetric difference, defined for sets A, B as

For example, the symmetric difference of {7, 8, 9, 10} and {9, 10, 11, 12} is the set {7, 8, 11, 12}. The power set of any set becomes a Boolean ring with symmetric difference as the addition of the ring (with the empty set as neutral element) and intersection as the multiplication of the ring.

Cartesian product

A new set can be constructed by associating every element of one set with every element of another set. The Cartesian product of two sets A and B, denoted by A × B is the set of all ordered pairs (a, b) such that a is a member of A and b is a member of B.

Examples:

  • {1, 2} × {red, white, green} = {(1, red), (1, white), (1, green), (2, red), (2, white), (2, green)}.
  • {1, 2} × {1, 2} = {(1, 1), (1, 2), (2, 1), (2, 2)}.
  • {a, b, c} × {d, e, f} = {(a, d), (a, e), (a, f), (b, d), (b, e), (b, f), (c, d), (c, e), (c, f)}.

Some basic properties of Cartesian products:

  • A × = ∅.
  • A × (BC) = (A × B) ∪ (A × C).
  • (AB) × C = (A × C) ∪ (B × C).

Let A and B be finite sets; then the cardinality of the Cartesian product is the product of the cardinalities:

  • | A × B | = | B × A | = | A | × | B |.

Applications

Set theory is seen as the foundation from which virtually all of mathematics can be derived. For example, structures in abstract algebra, such as groups, fields and rings, are sets closed under one or more operations.

One of the main applications of naive set theory is constructing relations. A relation from a domain A to a codomain B is a subset of the Cartesian product A × B. Given this concept, we are quick to see that the set F of all ordered pairs (x, x2), where x is real, is quite familiar. It has a domain set R and a codomain set that is also R, because the set of all squares is subset of the set of all real numbers. If placed in functional notation, this relation becomes f(x) = x2. The reason these two are equivalent is for any given value, y that the function is defined for, its corresponding ordered pair, (y, y2) is a member of the set F.

Axiomatic set theory

Although initially naive set theory, which defines a set merely as any well-defined collection, was well accepted, it soon ran into several obstacles. It was found that this definition spawned several paradoxes, most notably:

  • Russell's paradox – It shows that the "set of all sets that do not contain themselves," i.e. the "set" {x|x is a set and xx} does not exist.
  • Cantor's paradox – It shows that "the set of all sets" cannot exist.

The reason is that the phrase well-defined is not very well-defined. It was important to free set theory of these paradoxes because nearly all of mathematics was being redefined in terms of set theory. In an attempt to avoid these paradoxes, set theory was axiomatized based on first-order logic, and thus axiomatic set theory was born.

For most purposes, however, naive set theory is still useful.

Principle of inclusion and exclusion

A union B
The inclusion-exclusion principle can be used to calculate the size of the union of sets: the size of the union is the size of the two sets, minus the size of their intersection.

The inclusion–exclusion principle is a counting technique that can be used to count the number of elements in a union of two sets, if the size of each set and the size of their intersection are known. It can be expressed symbolically as

A more general form of the principle can be used to find the cardinality of any finite union of sets:

De Morgan's laws

Augustus De Morgan stated two laws about sets.

If A and B are any two sets then,

  • (A ∪ B)′ = A′ ∩ B′

The complement of A union B equals the complement of A intersected with the complement of B.

  • (A ∩ B)′ = A′ ∪ B′

The complement of A intersected with B is equal to the complement of A union to the complement of B.

See also

Notes

  1. ^ "Eine Menge, ist die Zusammenfassung bestimmter, wohlunterschiedener Objekte unserer Anschauung oder unseres Denkens – welche Elemente der Menge genannt werden – zu einem Ganzen." "Archived copy". Archived from the original on 2011-06-10. Retrieved 2011-04-22.CS1 maint: Archived copy as title (link)
  2. ^ a b Stoll, Robert. Sets, Logic and Axiomatic Theories. W. H. Freeman and Company. p. 5.

References

External links

Cantor's "Beiträge zur Begründung der transfiniten Mengenlehre" (in German)

Axiom schema

In mathematical logic, an axiom schema (plural: axiom schemata or axiom schemas) generalizes the notion of axiom.

Derived set (mathematics)

In mathematics, more specifically in point-set topology, the derived set of a subset S of a topological space is the set of all limit points of S. It is usually denoted by .

The concept was first introduced by Georg Cantor in 1872 and he developed set theory in large part to study derived sets on the real line.

Disjoint union

In set theory, the disjoint union (or discriminated union) of a family of sets is a modified union operation that indexes the elements according to which set they originated in. Or slightly different from this, the disjoint union of a family of subsets is the usual union of the subsets which are pairwise disjoint – disjoint sets means they have no element in common.

Note that these two concepts are different but strongly related. Moreover, it seems that they are essentially identical to each other in category theory. That is, both are realizations of the coproduct of category of sets.

Element (mathematics)

In mathematics, an element, or member, of a set is any one of the distinct objects that make up that set.

Hereditarily finite set

In mathematics and set theory, hereditarily finite sets are defined as finite sets whose elements are all hereditarily finite sets.

Infinite set

In set theory, an infinite set is a set that is not a finite set. Infinite sets may be countable or uncountable. Some examples are:

the set of all integers, {..., -1, 0, 1, 2, ...}, is a countably infinite set; and

the set of all real numbers is an uncountably infinite set.

Intersection (set theory)

In mathematics, the intersection A ∩ B of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently, all elements of B that also belong to A), but no other elements.For explanation of the symbols used in this article, refer to the table of mathematical symbols.

K-set (geometry)

In discrete geometry, a k-set of a finite point set S in the Euclidean plane is a subset of k elements of S that can be strictly separated from the remaining points by a line. More generally, in Euclidean space of higher dimensions, a k-set of a finite point set is a subset of k elements that can be separated from the remaining points by a hyperplane. In particular, when k = n/2 (where n is the size of S), the line or hyperplane that separates a k-set from the rest of S is a halving line or halving plane.

K-sets are related by projective duality to k-levels in line arrangements; the k-level in an arrangement of n lines in the plane is the curve consisting of the points that lie on one of the lines and have exactly k lines below them. Discrete and computational geometers have also studied levels in arrangements of more general kinds of curves and surfaces.

List of mathematical logic topics

This is a list of mathematical logic topics, by Wikipedia page.

For traditional syllogistic logic, see the list of topics in logic. See also the list of computability and complexity topics for more theory of algorithms.

List of set theory topics

This page is a list of articles related to set theory.

Recursive set

In computability theory, a set of natural numbers is called recursive, computable or decidable if there is an algorithm which takes a number as input, terminates after a finite amount of time (possibly depending on the given number) and correctly decides whether the number belongs to the set or not.

A more general class of sets consists of the recursively enumerable sets, also called semidecidable sets. For these sets, it is only required that there is an algorithm that correctly decides when a number is in the set; the algorithm may give no answer (but not the wrong answer) for numbers not in the set.

A set which is not computable is called noncomputable or undecidable.

Subset

In mathematics, a set A is a subset of a set B, or equivalently B is a superset of A, if A is "contained" inside B, that is, all elements of A are also elements of B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment. A is a subset of B may also be expressed as B includes A; or A is included in B.

The subset relation defines a partial order on sets.

The algebra of subsets forms a Boolean algebra in which the subset relation is called inclusion.

Thomas Jech

Thomas J. Jech (Czech: Tomáš Jech, pronounced [ˈtɔmaːʃ ˈjɛx]; born January 29, 1944 in Prague) is a mathematician specializing in set theory who was at Penn State for more than 25 years.

Type-2 fuzzy sets and systems

Type-2 fuzzy sets and systems generalize standard Type-1 fuzzy sets and systems so that more uncertainty can be handled. From the very beginning of fuzzy sets, criticism was made about the fact that the membership function of a type-1 fuzzy set has no uncertainty associated with it, something that seems to contradict the word fuzzy, since that word has the connotation of lots of uncertainty. So, what does one do when there is uncertainty about the value of the membership function? The answer to this question was provided in 1975 by the inventor of fuzzy sets, Prof. Lotfi A. Zadeh, when he proposed more sophisticated kinds of fuzzy sets, the first of which he called a type-2 fuzzy set. A type-2 fuzzy set lets us incorporate uncertainty about the membership function into fuzzy set theory, and is a way to address the above criticism of type-1 fuzzy sets head-on. And, if there is no uncertainty, then a type-2 fuzzy set reduces to a type-1 fuzzy set, which is analogous to probability reducing to determinism when unpredictability vanishes.

In order to symbolically distinguish between a type-1 fuzzy set and a type-2 fuzzy set, a tilde symbol is put over the symbol for the fuzzy set; so, A denotes a type-1 fuzzy set, whereas à denotes the comparable type-2 fuzzy set. When the latter is done, the resulting type-2 fuzzy set is called a general type-2 fuzzy set (to distinguish it from the special interval type-2 fuzzy set).

Prof. Zadeh didn't stop with type-2 fuzzy sets, because in that 1976 paper he also generalized all of this to type-n fuzzy sets. The present article focuses only on type-2 fuzzy sets because they are the next step in the logical progression from type-1 to type-n fuzzy sets, where n = 1, 2, … . Although some researchers are beginning to explore higher than type-2 fuzzy sets, as of early 2009, this work is in its infancy.

The membership function of a general type-2 fuzzy set, Ã, is three-dimensional (Fig. 1), where the third dimension is the value of the membership function at each point on its two-dimensional domain that is called its footprint of uncertainty (FOU).

For an interval type-2 fuzzy set that third-dimension value is the same (e.g., 1) everywhere, which means that no new information is contained in the third dimension of an interval type-2 fuzzy set. So, for such a set, the third dimension is ignored, and only the FOU is used to describe it. It is for this reason that an interval type-2 fuzzy set is sometimes called a first-order uncertainty fuzzy set model, whereas a general type-2 fuzzy set (with its useful third-dimension) is sometimes referred to as a second-order uncertainty fuzzy set model.

The FOU represents the blurring of a type-1 membership function, and is completely described by its two bounding functions (Fig. 2), a lower membership function (LMF) and an upper membership function (UMF), both of which are type-1 fuzzy sets! Consequently, it is possible to use type-1 fuzzy set mathematics to characterize and work with interval type-2 fuzzy sets. This means that engineers and scientists who already know type-1 fuzzy sets will not have to invest a lot of time learning about general type-2 fuzzy set mathematics in order to understand and use interval type-2 fuzzy sets.

Work on type-2 fuzzy sets languished during the 1980s and early-to-mid 1990's, although a small number of articles were published about them. People were still trying to figure out what to do with type-1 fuzzy sets, so even though Zadeh proposed type-2 fuzzy sets in 1976, the time was not right for researchers to drop what they were doing with type-1 fuzzy sets to focus on type-2 fuzzy sets. This changed in the latter part of the 1990s as a result of Prof. Jerry Mendel and his student's works on type-2 fuzzy sets and systems. Since then, more and more researchers around the world are writing articles about type-2 fuzzy sets and systems.

Uncountable set

In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal number is larger than that of the set of all natural numbers.

Union (set theory)

In set theory, the union (denoted by ∪) of a collection of sets is the set of all elements in the collection. It is one of the fundamental operations through which sets can be combined and related to each other.

For explanation of the symbols used in this article, refer to the table of mathematical symbols.

Universal set

In set theory, a universal set is a set which contains all objects, including itself. In set theory as usually formulated, the conception of a universal set leads to a paradox (Russell's paradox) and is consequently not allowed. However, some non-standard variants of set theory include a universal set.

General
Systems
Traditional logic
Predicate logic
Naive set theory
Set theory
Model theory
Proof theory
Computability
theory
Overview
Axioms
Operations
  • Concepts
  • Methods
Set types
Theories
Set theorists

This page is based on a Wikipedia article written by authors (here).
Text is available under the CC BY-SA 3.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.