In mathematics, a set is a collection of elements. [ 1 ] [ 2 ] [ 3 ] The elements that make up a rig can be any kind of mathematical objects : numbers, symbols, points in space, lines, other geometric shapes, variables, or even other sets. The rig with no chemical element is the vacate set ; a hardening with a individual element is a singleton. A jell may have a finite number of elements or be an space located. Two sets are equal if and only if they have precisely the same elements. [ 5 ] Sets are omnipresent 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 beginning half of the twentieth century.
Reading: Set (mathematics)
beginning [edit ]
The concept of a fix emerged in mathematics at the end of the nineteenth hundred. [ 6 ] The german parole for plant, Menge, was coined by Bernard Bolzano in his work Paradoxes of the Infinite. [ 7 ] [ 8 ] [ 9 ]
Menge for set is translated with aggregate here. passage with a translation of the master set definition of Georg Cantor. The german wordforis translated withhere. Georg Cantor, one of the founders of fixed theory, gave the adopt definition at the begin of his Beiträge zur Begründung der transfiniten Mengenlehre : [ 10 ]
A set is a gathering together into a wholly of definite, distinct objects of our percept or our thought—which are called elements of the sic .
Bertrand Russell called a fructify a class : “ When mathematicians deal with what they call a manifold, aggregate, Menge, ensemble, or some equivalent name, it is coarse, particularly where the number of terms involved is finite, to regard the aim in question ( which is in fact a class ) as defined by the count of its terms, and as consisting possibly of a single term, which is in that case is the class. ” [ 11 ]
Naïve set theory [edit ]
The foremost place of a put is that it can have elements, besides called members. Two sets are equal when they have the same elements. More precisely, sets A and B are adequate if every chemical element of A is a member of B, and every element of B is an chemical element of A ; this property is called the extensionality of sets. The simple concept of a set has proved enormously utilitarian 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 x ∉ x}, cannot exist.
- Cantor’s paradox shows that “the set of all sets” cannot exist.
Naïve set theory defines a fructify as any well-defined collection of distinct elements, but problems arise from the vagueness of the term well-defined .
axiomatic sic hypothesis [edit ]
In subsequent efforts to resolve these paradoxes since the time of the original formulation of naïve stage set hypothesis, the properties of sets have been defined by axioms. axiomatic set hypothesis takes the concept of a fixed as a primitive impression. [ 13 ] 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 hypothesis is free from paradox. [ citation needed ]
How sets are defined and set notation [edit ]
mathematical textbook normally denote sets by capital letters [ 14 ] in italic, such as A, B, C. [ 15 ] A set may besides be called a collection or family, specially when its elements are themselves sets .
Roster notation [edit ]
Roster or enumeration notation defines a set by listing its elements between curly brackets, separated by comma : [ 16 ] [ 17 ] [ 18 ] [ 19 ]
- A = { 4, 2, 1, 3 }
- B = { blue, whiten, crimson }.
In a fructify, all that matters is whether each element is in it or not, so the order of the elements in roll notation is irrelevant ( in contrast, in a sequence, a tuple, or a permutation of a set, the rate of the terms matters ). For exercise, { 2, 4, 6 } and { 4, 6, 4, 2 } represent the same rig. [ 20 ] [ 15 ] [ 21 ] For sets with many elements, specially those following an implicit pattern, the list of members can be abbreviated using an ellipsis ‘ … ‘. [ 22 ] [ 23 ] For exemplify, the determine of the first base thousand positive integers may be specified in roll notation as
- { 1, 2, 3, …, 1000 }.
Infinite sets in roll note [edit ]
An infinite set is a set with an endless list of elements. To describe an infinite set in roll note, an ellipsis is placed at the end of the list, or at both ends, to indicate that the list continues forever. For exemplar, the hardening of nonnegative integers is
-
{0, 1, 2, 3, 4, …}
,
and the located of all integers is
-
{…, −3, −2, −1, 0, 1, 2, 3, …}
.
semantic definition [edit ]
Another direction 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 bel be the set of colors of the French flag.
such a definition is called a semantic description. [ 25 ]
Set-builder note [edit ]
Set-builder note specifies a set as a excerpt from a larger set, determined by a condition on the elements. [ 25 ] [ 26 ] [ 27 ] For example, a set F can be defined as follows :
- farad = { normality ∣ nitrogen is an integer, and 0 ≤ n ≤ 19 }. { \displaystyle =\ { n\mid newton { \text { is an integer, and } } 0\leq n\leq 19\ }. }
In this notation, the erect banish “ | ” means “ such that ”, and the description can be interpreted as “ F is the set up of all numbers n such that nitrogen is an integer in the range from 0 to 19 inclusive ”. Some authors use a colon “ : ” rather of the vertical legal profession. [ 28 ]
Classifying methods of definition [edit ]
doctrine 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.[25] 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 [edit ]
If B is a fixed and ten is an element of B, this is written in shorthand as x ∈ B, which can besides be read as “ x belongs to B “, or “ x is in B “. The statement “ y is not an element of B “ is written as y ∉ B, which can besides be read as or “ y is not in B “. [ 29 ] [ 30 ] For case, with deference to the sets A = { 1, 2, 3, 4 }, B = { blue, white, crimson }, and F = { n | n is an integer, and 0 ≤ n ≤ 19 } ,
- 4 ∈ A and 12 ∈ F; and
- 20 ∉ F and k ∉ B.
The empty fit [edit ]
The empty set ( or null set ) is the unique set that has no members. It is denoted ∅ or ∅ { \displaystyle \emptyset } 
Singleton sets [edit ]
A singleton set is a located with precisely one component ; such a set may besides be called a unit set. [ 5 ] Any such set can be written as { x }, where x is the chemical element. The set { x } and the chemical element x intend different things ; Halmos draws the doctrine of analogy that a box containing a hat is not the lapp as the hat .
Subsets [edit ]
If every component of dress A is besides in B, then A is described as being a subset of B, or contained in B, written A ⊆ B, [ 36 ] or B ⊇ A. [ 37 ] 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 early : A ⊆ B and B ⊆ A 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 A ⊊ B. Likewise, B ⊋ A means B is a proper superset of A, i.e. B contains A, and is not equal to A. A third copulate of operators ⊂ and ⊃ are used differently by different authors : some authors use A ⊂ B and B ⊃ A to mean A is any subset of B ( and not necessarily a proper subset ), [ 29 ] while others reserve A ⊂ B and B ⊃ A 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 arrange is a subset of every hardened, and every located is a subset of itself :
- ∅ ⊆ A.
- A ⊆ A.
Euler and Venn diagrams [edit ]
A is a subset of B.
B is a superset of A. is a subset ofis a superset of An Euler diagram is a graphic representation of a collection of sets ; each set is depicted as a planar area enclosed by a closed circuit, with its elements inside. If A is a subset of B, then the region representing A is wholly inside the region representing B. If two sets have no elements in coarse, the regions do not overlap. A Venn diagram, in contrast, is a graphic representation of newton sets in which the nitrogen loops divide the plane into 2 n 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 exercise, 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 ( flush if such elements do not exist ) .
particular sets of numbers in mathematics [edit ]
There are sets of such mathematical importance, to which mathematicians refer thus frequently, that they have acquired limited names and notational conventions to identify them. many of these significant sets are represented in mathematical textbook using boldface ( e.g. Z { \displaystyle { \mathbf { Z } } } 
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, Q + { \displaystyle \mathbf { Q } ^ { + } } 
Functions [edit ]
A function ( or mapping ) from a rig A to a set B is a predominate that assigns to each “ input ” element of A an “ output signal ” that is an chemical element of B ; more formally, a serve 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 barn, 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 bacillus, 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 officiate is called a surjection, and a bijective serve is called a bijection or one-to-one correspondence .
cardinality [edit ]
The cardinality of a sic S, denoted | S |, is the number of members of S. [ 40 ] For example, if B = { blue, white, red }, then |B| = 3. Repeated members in roll notation are not counted, [ 41 ] [ 42 ] sol | { bluing, white, crimson, blue, white } | = 3, excessively. More formally, two sets parcel the lapp cardinality if there exists a one-to-one symmetry between them. The cardinality of the vacate set is zero. [ 43 ]
Infinite sets and space cardinality [edit ]
The number of elements of some sets is endless, or infinite. For example, the set N { \displaystyle \mathbb { N } } 
The Continuum Hypothesis [edit ]
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 lifelike numbers and the cardinality of a neat line. [ 46 ] In 1963, Paul Cohen proved that the Continuum Hypothesis is independent of the maxim system ZFC consist of Zermelo–Fraenkel set hypothesis with the axiom of choice. [ 47 ] ( ZFC is the most widely-studied adaptation of axiomatic set theory. )
Power sets [edit ]
The baron bent of a located S is the set of all subsets of S. [ 26 ] The empty sic and S itself are elements of the power set of S, because these are both subsets of S. For example, the office set of { 1, 2, 3 } is { ∅, { 1 }, { 2 }, { 3 }, { 1, 2 }, { 1, 3 }, { 2, 3 }, { 1, 2, 3 } }. The exponent put of a set S is normally written as P ( S ) or 2 S. [ 26 ] [ 15 ] If S has n elements, then P ( S ) has 2 n elements. For case, { 1, 2, 3 } has three elements, and its ability set has 23 = 8 elements, as shown above. If S is space ( whether countable or uncountable ), then P ( S ) is uncountable. furthermore, the power set is constantly strictly “ bigger ” than the original laid, in the sense that any attack to pair up the elements of S with the elements of P ( S ) will leave some elements of P ( S ) unpaired. ( There is never a bijection from S onto P ( S ). ) [ 50 ]
Partitions [edit ]
A partition of a set S is a located of nonempty subsets of S, such that every element x in S is in precisely one of these subsets. That is, the subsets are pairwise disjoin ( meaning any two sets of the partition hold no element in common ), and the marriage of all the subsets of the partition is S. [ 51 ]
basic operations [edit ]
There are respective fundamental operations for constructing newfangled sets from given sets .
Unions [edit ]
The union of
A
and
B
, denoted
A ∪ B
Two sets can be joined : the union of A and B, denoted by A ∪ B, is the hardened of all things that are members of A or of B or of both. Examples :
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- { 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:
- A ∪ B = B ∪ A .
- A ∪ ( B ∪ C ) = ( A ∪ B ) ∪ C .
- A ⊆ ( A ∪ B ) .
- A ∪ A = A .
- A ∪ ∅ = A .
- A ⊆ B if and only if A ∪ B = B .
Intersections [edit ]
A new set can besides be constructed by determining which members two sets have “ in coarse ”. The intersection of A and B, denoted by A ∩ B, is the determined of all things that are members of both A and B. If A ∩ B = ∅, then A and B are said to be disjoint .
The intersection of A and B, denoted
A ∩ B.
Examples :
- { 1, 2 } ∩ { 1, 2 } = { 1, 2 } .
- { 1, 2 } ∩ { 2, 3 } = { 2 } .
- { 1, 2 } ∩ { 3, 4 } = ∅ .
Some basic properties of intersections:
- A ∩ B = B ∩ A .
- A ∩ ( B ∩ C ) = ( A ∩ B ) ∩ C .
- A ∩ B ⊆ A .
- A ∩ A = A .
- A ∩ ∅ = ∅ .
- A ⊆ B if and only if A ∩ B = A .
Complements [edit ]
The relative complement
of B in A
The complement of A in U
The symmetric difference of A and B Two sets can besides be “ subtracted ”. The relative complement of B in A ( besides called the set-theoretic difference of A and B ), denoted by A \ B ( or A − B ), is the fixed 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 component green from the laid { 1, 2, 3 } ; doing thus 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 plainly complement of A, and is denoted by A ′ or Ac .
- 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 follow :
- A \ B ≠ B \ A for A ≠ B.
- A ∪ A ′ = U .
- A ∩ A ′ = ∅ .
- ( A ′ ) ′ = A .
- ∅ \ A = ∅ .
- A \ ∅ = A .
- A \ A = ∅ .
- A \ U = ∅ .
- A \ A ′ = A and A ′ \ A = A ′ .
- U ′ = ∅ and ∅′ = U .
- A \ B = A ∩ B ′.
- if A ⊆ B then A \ B = ∅ .
An propagation of the complement is the symmetrical dispute, defined for sets A, B as
- A Δ B = ( A ∖ B ) ∪ ( B ∖ A ). { \displaystyle A\, \Delta \, B= ( A\setminus B ) \cup ( B\setminus A ). }
For example, the symmetrical difference of { 7, 8, 9, 10 } and { 9, 10, 11, 12 } is the place { 7, 8, 11, 12 }. The power set of any fructify becomes a boolean ring with symmetrical difference as the addition of the ring ( with the empty set as impersonal component ) and intersection as the multiplication of the closed chain .
cartesian product [edit ]
A new fit can be constructed by associating every element of one set with every component of another stage set. The Cartesian product of two sets A and B, denoted by A × B, is the set up of all ordered pairs ( a, b ) such that a is a penis of A and b is a member of B. Examples :
- { 1, 2 } × { red, white, green } = { ( 1, loss ), ( 1, blank ), ( 1, green ), ( 2, crimson ), ( 2, white ), ( 2, fleeceable ) } .
- { 1, 2 } × { 1, 2 } = { ( 1, 1 ), ( 1, 2 ), ( 2, 1 ), ( 2, 2 ) } .
- { a, b, hundred } × { five hundred, vitamin e, degree fahrenheit } = { ( a, vitamin d ), ( a, einsteinium ), ( a, degree fahrenheit ), ( b, d ), ( bel, east ), ( b-complex vitamin, f ), ( c, five hundred ), ( c, e ), ( c, farad ) } .
Some basic properties of cartesian products :
- A × ∅ = ∅.
- A × ( B ∪ C ) = ( A × B ) ∪ ( A × C ) .
- ( A ∪ B ) × C = ( A × C ) ∪ ( B × C ) .
Let A and B be finite sets ; then the cardinality of the cartesian intersection is the intersection of the cardinalities :
- |A × B| = |B × A| = |A| × |B|.
Applications [edit ]
Sets are omnipresent in modern mathematics. For case, structures in abstract algebra, such as groups, fields and rings, are sets closed under one or more operations. One of the chief applications of naive specify theory is in the construction of relations. A relation back from a domain A to a codomain B is a subset of the cartesian product A × B. For example, considering the adjust S = { rock, paper, scissors } of shapes in the bet on of the lapp name, the relation “ beats ” from S to S is the determined B = { ( scissors, paper ), ( composition, rock ), ( rock ‘n’ roll, scissors ) } ; therefore x beats y in the plot if the pair ( x, y ) is a member of B. Another exemplar is the fit F of all pairs ( x, x 2 ), 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 number numbers. Since for every x in R, one and lone one pair ( x, … ) is found in F, it is called a function. In functional notation, this relation can be written as F ( x ) = x 2 .
rationale of inclusion and excommunication [edit ]
The inclusion-exclusion principle is 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 count technique that can be used to count the numeral of elements in a union of two sets—if the size of each fixed and the size of their overlap are known. It can be expressed symbolically as
- | A ∪ B | = | A | + | B | − | A ∩ B |. { \displaystyle |A\cup B|=|A|+|B|-|A\cap B|. }
A more general form of the rationale can be used to find the cardinality of any finite union of sets :
- | A 1 ∪ A 2 ∪ A 3 ∪ … ∪ A n | = ( | A 1 | + | A 2 | + | A 3 | + … | A newton | ) − ( | A 1 ∩ A 2 | + | A 1 ∩ A 3 | + … | A nitrogen − 1 ∩ A north | ) + … + ( − 1 ) normality − 1 ( | A 1 ∩ A 2 ∩ A 3 ∩ … ∩ A n | ). { \displaystyle { \begin { aligned } \left|A_ { 1 } \cup A_ { 2 } \cup A_ { 3 } \cup \ldots \cup A_ { north } \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_ { north } \right|\right ) \\ & { } +\ldots \\ & { } +\left ( -1\right ) ^ { n-1 } \left ( \left|A_ { 1 } \cap A_ { 2 } \cap A_ { 3 } \cap \ldots \cap A_ { nitrogen } \right|\right ) .\end { aligned } } }
De Morgan ‘s laws [edit ]
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 coupling B equals the complement of A intersect with the complement of B .
- ( A ∩ B ) ′ = A ′ ∪ B ′
The complement of A intersect with B is equal to the complement of A union to the complement of B .