Set (mathematics)

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In mathematics a set is a collection of distinct elements. [1] [2] [3] The elements that make up a set can be any kind of things: people, letters of the alphabet, numbers, points in space, lines, other geometrical shapes, variables, or even other sets. [4] Two sets are equal if and only if they have precisely the same elements. [5]

Contents

Sets are ubiquitous in modern mathematics. Indeed, set theory, more specifically Zermelo–Fraenkel set theory, has been the standard way to provide rigorous foundations for all branches of mathematics since the first half of the 20th century. [4]

Origin

The concept of a set emerged in mathematics at the end of the 19th century. [6] The German word for set, Menge, was coined by Bernard Bolzano in his work Paradoxes of the Infinite . [7] [8] [9]

Georg Cantor, one of the founders of set theory, gave the following definition at the beginning of his Beiträge zur Begründung der transfiniten Mengenlehre: [10]

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

Naïve set theory

The foremost property of a set is that it can have elements, also called members. Two sets are equal when they have the same elements. More precisely, sets A and B are equal if every element of A is a member of B, and every element of B is an element of A; this property is called the extensionality of sets. [11]

The simple concept of a set has proved enormously useful in mathematics, but paradoxes arise if no restrictions are placed on how sets can be constructed:

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

Naïve set theory defines a set as any well-defined collection of distinct elements, but problems arise from the vagueness of the term well-defined.

Axiomatic set theory

In subsequent efforts to resolve these paradoxes since the time of the original formulation of naïve set theory, the properties of sets have been defined by axioms. Axiomatic set theory takes the concept of a set as a primitive notion. [12] The purpose of the axioms is to provide a basic framework from which to deduce the truth or falsity of particular mathematical propositions (statements) about sets, using first-order logic. According to Gödel's incompleteness theorems however, it is not possible to use first-order logic to prove any such particular axiomatic set theory is free from paradox.[ citation needed ]

How sets are defined and set notation

Mathematical texts commonly denote sets by capital letters [13] [4] [14] in italic, such as A, B, C. [14] [15] A set may also be called a collection or family, especially when its elements are themselves sets.

Semantic definition

One way to define a set is to use a rule to determine what the elements are:

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

Such a definition is also called a semantic description. [16] [17]

Roster notation

Roster or enumeration notation defines a set by listing its elements between curly brackets, separated by commas: [18] [19] [20] [21]

A = {4, 2, 1, 3}
B = {blue, white, red}.

In a set, all that matters is whether each element is in it or not, so the ordering of the elements in roster notation is irrelevant (in contrast, in a sequence, a tuple, or a permutation of a set, the ordering of the terms matters). For example, {2, 4, 6} and {4, 6, 2} represent the same set. [22] [15] [23]

For sets with many elements, especially those following an implicit pattern, the list of members can be abbreviated using an ellipsis’. [24] [25] For instance, the set of the first thousand positive integers may be specified in roster notation as

{1, 2, 3, …, 1000}.

Infinite sets in roster notation

An infinite set is a set with an endless list of elements. To describe an infinite set in roster notation, an ellipsis is placed at the end of the list, or at both ends, to indicate that the list continues forever. For example, the set of nonnegative integers is

{0, 1, 2, 3, 4, …},

and the set of all integers is

{…, −3, −2, −1, 0, 1, 2, 3, …}.

Set-builder notation

Set-builder notation specifies a set as a selection from a larger set, determined by a condition on the elements. [17] [26] [27] For example, a set F can be defined as follows:

F${\displaystyle =\{n\mid n{\text{ is an integer, and }}0\leq n\leq 19\}.}$

In this notation, the vertical bar "|" means "such that", and the description can be interpreted as "F is the set of all numbers n such that n is an integer in the range from 0 to 19 inclusive". Some authors use a colon ":" instead of the vertical bar. [28]

Classifying methods of definition

Philosophy uses specific terms to classify types of definitions:

• An intensional definition uses a rule to determine membership. Semantic definitions and definitions using set-builder notation are examples.
• An extensional definition describes a set by listing all its elements. [17] Such definitions are also called enumerative .
• An ostensive definition is one that describes a set by giving examples of elements; a roster involving an ellipsis would be an example.

Membership

If B is a set and x is an element of B, this is written in shorthand as xB, which can also be read as “x belongs to B”, or “x is in B”. [11] The statement “y is not an element of B” is written as yB, which can also be read as or “y is not in B”. [29] [14] [30]

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

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

The empty set

The empty set (or null set) is the unique set that has no members. It is denoted or ${\displaystyle \emptyset }$ or { } [31] [14] [32] or ϕ [33] (or ϕ). [34]

Singleton sets

A singleton set is a set with exactly one element; such a set may also be called a unit set. [5] Any such set can be written as {x}, where x is the element. The set {x} and the element x mean different things; Halmos [35] draws the analogy that a box containing a hat is not the same as the hat.

Subsets

If every element of set A is also in B, then A is described as being a subset of B, or contained in B, written AB, [36] or BA. [37] [14] The latter notation may be read B contains A, B includes A, or B is a superset of A. The relationship between sets established by ⊆ is called inclusion or containment. Two sets are equal if they contain each other: AB and BA is equivalent to A = B. [26]

If A is a subset of B, but A is not equal to B, then A is called a proper subset of B. This can be written AB. Likewise, BA means B is a proper superset of A, i.e. B contains A, and is not equal to A.

A third pair of operators ⊂ and ⊃ are used differently by different authors: some authors use AB and BA to mean A is any subset of B (and not necessarily a proper subset), [38] [29] while others reserve AB and BA for cases where A is a proper subset of B. [36]

Examples:

• The set of all humans is a proper subset of the set of all mammals.
• {1, 3}{1, 2, 3, 4}.
• {1, 2, 3, 4}{1, 2, 3, 4}.

The empty set is a subset of every set, [31] and every set is a subset of itself: [38]

• ∅ ⊆ A.
• AA.

Euler and Venn diagrams

An Euler diagram is a graphical representation of a collection of sets; each set is depicted as a planar region enclosed by a loop, with its elements inside. If A is a subset of B, then the region representing A is completely inside the region representing B. If two sets have no elements in common, the regions do not overlap.

A Venn diagram, in contrast, is a graphical representation of n sets in which the n loops divide the plane into 2n zones such that for each way of selecting some of the n sets (possibly all or none), there is a zone for the elements that belong to all the selected sets and none of the others. For example, if the sets are A, B, and C, there should be a zone for the elements that are inside A and C and outside B (even if such elements do not exist).

Special sets of numbers in mathematics

There are sets of such mathematical importance, to which mathematicians refer so frequently, that they have acquired special names and notational conventions to identify them.

Many of these important sets are represented in mathematical texts using bold (e.g. ${\displaystyle {\mathbf {Z}}}$) or blackboard bold (e.g. ${\displaystyle \mathbb {Z} }$) typeface. [39] These include [14]

• ${\displaystyle {\mathbf {N}}}$ or ${\displaystyle \mathbb {N} }$, the set of all natural numbers: ${\displaystyle {\mathbf {N}}=\{0,1,2,3,...\}}$ (often, authors exclude 0); [39]
• ${\displaystyle {\mathbf {Z}}}$ or ${\displaystyle \mathbb {Z} }$, the set of all integers (whether positive, negative or zero): ${\displaystyle {\mathbf {Z}}=\{...,-2,-1,0,1,2,3,...\}}$; [39]
• ${\displaystyle {\mathbf {Q}}}$ or ${\displaystyle \mathbb {Q} }$, the set of all rational numbers (that is, the set of all proper and improper fractions): ${\displaystyle {\mathbf {Q}}=\left\{{\frac {a}{b}}\mid a,b\in {\mathbf {Z}},b\neq 0\right\}}$. For example, 7/4Q and 5 = 5/1Q; [39]
• ${\displaystyle {\mathbf {R}}}$ or ${\displaystyle \mathbb {R} }$, the set of all real numbers, including all rational numbers and all irrational numbers (which include algebraic numbers such as ${\displaystyle {\sqrt {2}}}$ that cannot be rewritten as fractions, as well as transcendental numbers such as π and e); [39]
• ${\displaystyle {\mathbf {C}}}$ or ${\displaystyle \mathbb {C} }$, the set of all complex numbers: C = {a + bi | a, bR}, for example, 1 + 2iC. [39]

Each of the above sets of numbers has an infinite number of elements. Each is a subset of the sets listed below it.

Sets of positive or negative numbers are sometimes denoted by superscript plus and minus signs, respectively. For example, ${\displaystyle \mathbf {Q} ^{+}}$ represents the set of positive rational numbers.

Functions

A function (or mapping ) from a set A to a set B is a rule that assigns to each "input" element of A an "output" that is an element of B; more formally, a function is a special kind of relation, one that relates each element of A to exactly one element of B. A function is called

• injective (or one-to-one) if it maps any two different elements of A to different elements of B,
• surjective (or onto) if for every element of B, there is at least one element of A that maps to it, and
• bijective (or a one-to-one correspondence) if the function is both injective and surjective — in this case, each element of A is paired with a unique element of B, and each element of B is paired with a unique element of A, so that there are no unpaired elements.

An injective function is called an injection, a surjective function is called a surjection, and a bijective function is called a bijection or one-to-one correspondence.

Cardinality

The cardinality of a set S, denoted |S|, is the number of members of S. [40] For example, if B = {blue, white, red}, then |B| = 3. Repeated members in roster notation are not counted, [41] [42] so |{blue, white, red, blue, white}| = 3, too.

More formally, two sets share the same cardinality if there exists a one-to-one correspondence between them.

The cardinality of the empty set is zero. [43]

Infinite sets and infinite cardinality

The list of elements of some sets is endless, or infinite . For example, the set ${\displaystyle \mathbb {N} }$ of natural numbers is infinite. [26] In fact, all the special sets of numbers mentioned in the section above, are infinite. Infinite sets have infinite cardinality.

Some infinite cardinalities are greater than others. Sets with the same cardinality as ${\displaystyle \mathbb {N} }$ are called countable sets . Arguably one of the most significant results from set theory is that the set of real numbers has greater cardinality than the set of natural numbers. [44] Sets with cardinality greater than the set of natural numbers are called uncountable sets.

However, it can be shown that the cardinality of a straight line (i.e., the number of points on a 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. [45]

The Continuum Hypothesis

The Continuum Hypothesis, formulated by Georg Cantor in 1878, is the statement that there is no set with cardinality strictly between the cardinality of the natural numbers and the cardinality of a straight line. [46] In 1963, Paul Cohen proved that the Continuum Hypothesis is independent of the axiom system ZFC consisting of Zermelo–Fraenkel set theory with the axiom of choice. [47] (ZFC is the most widely-studied version of axiomatic set theory.)

Power sets

The power set of a set S is the set of all subsets of S. [26] The empty set and S itself are elements of the power set of S because these are both subsets of S. For example, the power set of {1, 2, 3} is {∅, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}. The power set of a set S is commonly written as P(S) or 2P. [26] [48] [14] [15]

The power set of a finite set with n elements has 2n elements. [49] 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, within the most widely-used frameworks of set theory, 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).) [50]

Partitions

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, the subsets are pairwise disjoint (meaning any two sets of the partition contain no element in common), and the union of all the subsets of the partition is S. [51] [52]

Basic operations

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

Unions

Two sets can be joined: the union of A and B, denoted by AB, [14] is the set of all things that are members of A or of B or of both.

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.
• 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, [14] 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.

Examples:

• {1, 2} ∩ {1, 2} = {1, 2}.
• {1, 2} ∩ {2, 3} = {2}.
• {1, 2} ∩ {3, 4} = ∅.

Some basic properties of intersections:

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

Complements

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), [14] is the set of all elements that are members of A, but not members of B. 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 will not affect the elements in the set.

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′ or Ac. [14]

• 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 include the following:

• 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

${\displaystyle A\,\Delta \,B=(A\setminus B)\cup (B\setminus A).}$

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, [14] 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

Sets are ubiquitous in modern mathematics. 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 in the construction of relations. A relation from a domain A to a codomain B is a subset of the Cartesian product A × B. For example, considering the set S = {rock, paper, scissors} of shapes in the game of the same name, the relation “beats” from S to S is the set B = {(scissors,paper), (paper,rock), (rock,scissors)}; thus x beats y in the game if the pair (x,y) is a member of B. Another example is the set F of all pairs (x, x2), where x is real. This relation is a subset of R × R, because the set of all squares is subset of the set of all real numbers. Since for every x in R, one and only one pair (x,...) is found in F, it is called a function. In functional notation, this relation can be written as F(x) = x2.

Principle of inclusion and exclusion

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

${\displaystyle |A\cup B|=|A|+|B|-|A\cap B|.}$

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

{\displaystyle {\begin{aligned}\left|A_{1}\cup A_{2}\cup A_{3}\cup \ldots \cup A_{n}\right|=&\left(\left|A_{1}\right|+\left|A_{2}\right|+\left|A_{3}\right|+\ldots \left|A_{n}\right|\right)\\&{}-\left(\left|A_{1}\cap A_{2}\right|+\left|A_{1}\cap A_{3}\right|+\ldots \left|A_{n-1}\cap A_{n}\right|\right)\\&{}+\ldots \\&{}+\left(-1\right)^{n-1}\left(\left|A_{1}\cap A_{2}\cap A_{3}\cap \ldots \cap A_{n}\right|\right).\end{aligned}}}

De Morgan's laws

Augustus De Morgan stated two laws about sets.

If A and B are any two sets then,

• (AB)′ = A′ ∩ B

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

• (AB)′ = A′ ∪ B

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

Notes

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Naive set theory is any of several theories of sets used in the discussion of the foundations of mathematics. Unlike axiomatic set theories, which are defined using formal logic, naive set theory is defined informally, in natural language. It describes the aspects of mathematical sets familiar in discrete mathematics, and suffices for the everyday use of set theory concepts in contemporary mathematics.

In mathematics, a countable set is a set with the same cardinality as some subset of the set of natural numbers. A countable set is either a finite set or a countably infinite set. Whether finite or infinite, the elements of a countable set can always be counted one at a time and—although the counting may never finish—every element of the set is associated with a unique natural number.

In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. The transfinite cardinal numbers, often denoted using the Hebrew symbol (aleph) followed by a subscript, describe the sizes of infinite sets.

In mathematics, any vector space has a corresponding dual vector space consisting of all linear forms on , together with the vector space structure of pointwise addition and scalar multiplication by constants.

In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word isomorphism is derived from the Ancient Greek: ἴσος isos "equal", and μορφή morphe "form" or "shape".

In mathematics, a set B of vectors in a vector space V is called a basis if every element of V may be written in a unique way as a finite linear combination of elements of B. The coefficients of this linear combination are referred to as components or coordinates of the vector with respect to B. The elements of a basis are called basis vectors.

In mathematics, the natural numbers are those used for counting and ordering. In common mathematical terminology, words colloquially used for counting are "cardinal numbers", and words used for ordering are "ordinal numbers". The natural numbers can, at times, appear as a convenient set of codes, that is, as what linguists call nominal numbers, forgoing many or all of the properties of being a number in a mathematical sense. The set of natural numbers is often denoted by the symbol .

In mathematics, the power set of a set S is the set of all subsets of S, including the empty set and S itself. In axiomatic set theory, the existence of the power set of any set is postulated by the axiom of power set. The powerset of S is variously denoted as P(S), 𝒫(S), P(S), (S), ℘(S), or 2S. The notation 2S is used because given any set with exactly two elements, the powerset of S can be identified with the set of all functions from S into that set.

In set theory, the union 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. A nullary union refers to a union of zero sets and it is by definition equal to the empty set.

In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist. In other words, a ring is a set equipped with two binary operations satisfying properties analogous to those of addition and multiplication of integers. Ring elements may be numbers such as integers or complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series.

In mathematics, a (real) interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers x satisfying 0 ≤ x ≤ 1 is an interval which contains 0, 1, and all numbers in between. Other examples of intervals are the set of numbers such that 0 < x < 1, the set of all real numbers , the set of nonnegative real numbers, the set of positive real numbers, the empty set, and any singleton.

A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object, an action on mathematical objects, a relation between mathematical objects, or for structuring the other symbols that occur in a formula. As formulas are entirely constituted with symbols of various types, many symbols are needed for expressing all mathematics.

In mathematics, a function is a binary relation between two sets that associates each element of the first set to exactly one element of the second set. Typical examples are functions from integers to integers, or from the real numbers to real numbers.

In axiomatic set theory and the branches of mathematics and philosophy that use it, the axiom of infinity is one of the axioms of Zermelo–Fraenkel set theory. It guarantees the existence of at least one infinite set, namely a set containing the natural numbers. It was first published by Ernst Zermelo as part of his set theory in 1908.

In mathematics, a multiset is a modification of the concept of a set that, unlike a set, allows for multiple instances for each of its elements. The number of instances given for each element is called the multiplicity of that element in the multiset. As a consequence, an infinite number of multisets exist which contain only elements a and b, but vary in the multiplicities of their elements:

In elementary set theory, Cantor's theorem is a fundamental result which states that, for any set , the set of all subsets of has a strictly greater cardinality than itself. For finite sets, Cantor's theorem can be seen to be true by simple enumeration of the number of subsets. Counting the empty set as a subset, a set with members has a total of subsets, so that if then , and the theorem holds because for all non-negative integers.

In mathematics, the image of a function is the set of all output values it may produce.

In mathematics, an element of a set is any one of the distinct objects that belong to that set.

In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line. The adjective real in this context was introduced in the 17th century by René Descartes, who distinguished between real and imaginary roots of polynomials. The real numbers include all the rational numbers, such as the integer −5 and the fraction 4/3, and all the irrational numbers, such as 2. Included within the irrationals are the real transcendental numbers, such as π (3.14159265...). In addition to measuring distance, real numbers can be used to measure quantities such as time, mass, energy, velocity, and many more. The set of real numbers is denoted using the symbol R or and is sometimes called "the reals".