About Sets

7/27/2019 About Sets

1/54

SET

DEFINITION

A set is a collection of distinct objects, considered as an object in its own right.

Sets are one of the most fundamental concepts 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 such as Venn diagrams

are taught at a young age, while more advanced concepts are taught as part of a

university degree. By a "set" we mean any collection M into a whole of definite,

distinct objects m (which called the "elements" ofM).

The elements or members of a set can be anything: numbers, people, letters of

the alphabet, other sets, and so on. Sets are conventionally denoted with capital

letters. Sets A and B are equal if and only if they have precisely the same

elements.

As discussed below, the definition given above turned out to be inadequate for

formal mathematics; instead, the notion of a "set" is taken as an undefined

primitive in axiomatic set theory. The most basic properties are that a set "has"elements, and that two sets are equal. If and only if they have the same

elements. The intersection of two sets is made up of the objects contained in

both sets, shown in a Venn diagram:

There are two ways of describing, or specifying the members of, a set. One way

is by intensional definition, using a rule orsemantic description:
http://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/Mathematics_educationhttp://en.wikipedia.org/wiki/Venn_diagramhttp://en.wikipedia.org/wiki/Element_%28mathematics%29http://en.wikipedia.org/wiki/Capital_lettershttp://en.wikipedia.org/wiki/Capital_lettershttp://en.wikipedia.org/wiki/If_and_only_ifhttp://en.wikipedia.org/wiki/Undefined_primitivehttp://en.wikipedia.org/wiki/Undefined_primitivehttp://en.wikipedia.org/wiki/Axiomatic_set_theoryhttp://en.wikipedia.org/wiki/Venn_diagramhttp://en.wikipedia.org/wiki/Intensional_definitionhttp://en.wikipedia.org/wiki/Semantichttp://en.wikipedia.org/wiki/File:Venn_A_intersect_B.svghttp://en.wikipedia.org/wiki/Semantichttp://en.wikipedia.org/wiki/Intensional_definitionhttp://en.wikipedia.org/wiki/Venn_diagramhttp://en.wikipedia.org/wiki/Axiomatic_set_theoryhttp://en.wikipedia.org/wiki/Undefined_primitivehttp://en.wikipedia.org/wiki/Undefined_primitivehttp://en.wikipedia.org/wiki/If_and_only_ifhttp://en.wikipedia.org/wiki/Capital_lettershttp://en.wikipedia.org/wiki/Capital_lettershttp://en.wikipedia.org/wiki/Element_%28mathematics%29http://en.wikipedia.org/wiki/Venn_diagramhttp://en.wikipedia.org/wiki/Mathematics_educationhttp://en.wikipedia.org/wiki/Mathematics


2/54

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 brackets:

C= {4, 2, 1, 3}

D = {blue, white, red}

Unlike a multiset, every element of a set must be unique; no two members may

be identical. All set operations preserve the property that each element of the set

is unique. The order in which the elements of a set are listed is irrelevant, unlike

a sequence ortuple. For example,

{6, 11} = {11, 6} = {11, 11, 6, 11},

Because the extensional specification means merely that each of the elements

listed is a member of the 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 specifiedextensionally as:

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

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

Ellipses may also be used where sets have infinitely many members. Thus the

set of positive even numbers can be written as {2, 4, 6, 8, ... }.

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 , , , and . A more

general form of this is set-builder notation, through which, for instance, the set F
http://en.wikipedia.org/wiki/Integerhttp://en.wikipedia.org/wiki/Flag_of_Francehttp://en.wikipedia.org/wiki/Extension_%28semantics%29http://en.wikipedia.org/wiki/Extensional_definitionhttp://en.wikipedia.org/wiki/Brackethttp://en.wikipedia.org/wiki/Multisethttp://en.wikipedia.org/wiki/Set_operations_%28mathematics%29http://en.wikipedia.org/wiki/Sequencehttp://en.wikipedia.org/wiki/Tuplehttp://en.wikipedia.org/wiki/Ellipsishttp://en.wikipedia.org/wiki/Even_numberhttp://en.wikipedia.org/wiki/Set-builder_notationhttp://en.wikipedia.org/wiki/Set-builder_notationhttp://en.wikipedia.org/wiki/Even_numberhttp://en.wikipedia.org/wiki/Ellipsishttp://en.wikipedia.org/wiki/Tuplehttp://en.wikipedia.org/wiki/Sequencehttp://en.wikipedia.org/wiki/Set_operations_%28mathematics%29http://en.wikipedia.org/wiki/Multisethttp://en.wikipedia.org/wiki/Brackethttp://en.wikipedia.org/wiki/Extensional_definitionhttp://en.wikipedia.org/wiki/Extension_%28semantics%29http://en.wikipedia.org/wiki/Flag_of_Francehttp://en.wikipedia.org/wiki/Integer


3/54

of the twenty smallest integers that are four less than perfect squares can be

denoted:

F= {n2 4 : nis an integer; and 0 n 19}

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.

One often has the choice of specifying a set intensionally or extensionally. In the

examples above, for instance,A = Cand B = D.

Membership

Main article: Element (mathematics)

The key relation between sets is membership when one set is an element of

another. IfA is a member ofB, this is denotedAB, while ifCis not a member

ofB then CB. For example, with respect to the sets A = {1,2,3,4}, B = {blue,

white, red}, and F= {n2 4 : nis an integer; and 0 n 19} defined above,

4 A and 285 F; but

9 Fand green B.

Subsets

If every member of setA is also a member of set B, thenA is said to be a subset

ofB, writtenAB (also pronouncedA is contained in B). Equivalently, we can

write BA, read as B is a superset of A, B includes A, orB contains A. The

relationship between sets established by is called inclusion orcontainment.

IfA is a subset of, but not equal to, B, then A is called a proper subset ofB,

writtenAB (A is a proper subset of B) orBA (B is proper superset of A).
http://en.wikipedia.org/wiki/Square_numberhttp://en.wikipedia.org/wiki/Colon_%28punctuation%29http://en.wikipedia.org/wiki/Vertical_barhttp://en.wikipedia.org/wiki/Element_%28mathematics%29http://en.wikipedia.org/wiki/Relation_%28mathematics%29http://en.wikipedia.org/wiki/Relation_%28mathematics%29http://en.wikipedia.org/wiki/Element_%28mathematics%29http://en.wikipedia.org/wiki/Vertical_barhttp://en.wikipedia.org/wiki/Colon_%28punctuation%29http://en.wikipedia.org/wiki/Square_number


4/54

Note that the expressions A B and BA are used differently by different

authors; some authors use them to mean the same as AB (respectively B

A), whereas other use them to mean the same asAB (respectively BA).

A is a subset ofB

Example:

The set of all men is a propersubset 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.

An obvious but useful identity, which can often be used to show that two

seemingly different sets are equal:

A = B if and only ifAB and BA.

Power sets

The power set of a set S is the set of all subsets ofS. This includes the subsets

formed from all the members of S and the empty set. If a finite set S has

cardinality n then the power set of S has cardinality 2n. The power set can be

written as P(S).
http://en.wikipedia.org/wiki/Subsethttp://en.wikipedia.org/wiki/File:Venn_A_subset_B.svghttp://en.wikipedia.org/wiki/Subset


5/54

IfS is an infinite (eithercountable oruncountable) set then the power set ofS is

always uncountable. Moreover, ifS is a set, then there is never a bijection from S

onto P(S). In other words, the power set ofS is always strictly "bigger" than S.

As an example, the power set of {1, 2, 3} is {{1, 2, 3}, {1, 2}, {1, 3}, {2, 3}, {1}, {2},

{3}, }. The cardinality of the original set is 3, and the cardinality of the power set

is 23

= 8. This relationship is one of the reasons for the terminologypower set.

Cardinality

The cardinality |S| of a set S is "the number of members of S." For example,

since the French flag has three colors, |B| = 3.

There is a unique set with no members and zero cardinality, which is called the

empty set (or the null set) and is denoted by the symbol (other notations are

used; see empty set). 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 ofaxiomatic set theory.

Some sets have infinite cardinality. The set N ofnatural 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.
http://en.wikipedia.org/wiki/Countablehttp://en.wikipedia.org/wiki/Uncountablehttp://en.wikipedia.org/wiki/Bijectionhttp://en.wikipedia.org/wiki/Empty_sethttp://en.wikipedia.org/wiki/Empty_sethttp://en.wikipedia.org/wiki/Empty_sethttp://en.wikipedia.org/wiki/0_%28number%29http://en.wikipedia.org/wiki/Axiomatic_set_theoryhttp://en.wikipedia.org/wiki/Infinite_sethttp://en.wikipedia.org/wiki/Natural_numberhttp://en.wikipedia.org/wiki/Real_numberhttp://en.wikipedia.org/wiki/Straight_linehttp://en.wikipedia.org/wiki/Line_segmenthttp://en.wikipedia.org/wiki/Plane_%28mathematics%29http://en.wikipedia.org/wiki/Dimension_%28mathematics%29http://en.wikipedia.org/wiki/Euclidean_spacehttp://en.wikipedia.org/wiki/Euclidean_spacehttp://en.wikipedia.org/wiki/Dimension_%28mathematics%29http://en.wikipedia.org/wiki/Plane_%28mathematics%29http://en.wikipedia.org/wiki/Line_segmenthttp://en.wikipedia.org/wiki/Straight_linehttp://en.wikipedia.org/wiki/Real_numberhttp://en.wikipedia.org/wiki/Natural_numberhttp://en.wikipedia.org/wiki/Infinite_sethttp://en.wikipedia.org/wiki/Axiomatic_set_theoryhttp://en.wikipedia.org/wiki/0_%28number%29http://en.wikipedia.org/wiki/Empty_sethttp://en.wikipedia.org/wiki/Empty_sethttp://en.wikipedia.org/wiki/Bijectionhttp://en.wikipedia.org/wiki/Uncountablehttp://en.wikipedia.org/wiki/Countable


6/54

Special sets

There are some sets which 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. Many of these sets

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

include:

P, denoting the set of all primes:P = {2, 3, 5, 7, 11, 13, 17, ...}.

N, denoting the set of all natural numbers:N = {1, 2, 3, . . .}.

Z, denoting the set of all integers (whether positive, negative or zero): Z =

{... , 2, 1, 0, 1, 2, ...}. Q, denoting the set of all rational numbers (that is, the set of all properand

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.

R, denoting the set of all real numbers. This set includes all rational

numbers, together with all irrational numbers (that is, numbers which

cannot be rewritten as fractions, such as,e, and 2, as well asnumbers

that cannot be defined).

C, denoting the set of all complex numbers:C = {a + bi: a, bR}. For

example, 1 + 2iC.

H, denoting the set of all quaternions:H = {a + bi+ cj+ dk: a, b, c, dR}.

For example, 1 + i+ 2j kH.

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.
http://en.wikipedia.org/wiki/Blackboard_boldhttp://en.wikipedia.org/wiki/Prime_numberhttp://en.wikipedia.org/wiki/Natural_numberhttp://en.wikipedia.org/wiki/Integerhttp://en.wikipedia.org/wiki/Rational_numberhttp://en.wikipedia.org/wiki/Proper_fractionhttp://en.wikipedia.org/wiki/Improper_fractionhttp://en.wikipedia.org/wiki/Real_numberhttp://en.wikipedia.org/wiki/Irrational_numberhttp://en.wikipedia.org/wiki/Pihttp://en.wikipedia.org/wiki/Pihttp://en.wikipedia.org/wiki/Pihttp://en.wikipedia.org/wiki/Exponentiationhttp://en.wikipedia.org/wiki/Exponentiationhttp://en.wikipedia.org/wiki/Exponentiationhttp://en.wikipedia.org/wiki/Definable_real_numberhttp://en.wikipedia.org/wiki/Definable_real_numberhttp://en.wikipedia.org/wiki/Definable_real_numberhttp://en.wikipedia.org/wiki/Complex_numberhttp://en.wikipedia.org/wiki/Quaternionhttp://en.wikipedia.org/wiki/Number_theoryhttp://en.wikipedia.org/wiki/Number_theoryhttp://en.wikipedia.org/wiki/Quaternionhttp://en.wikipedia.org/wiki/Complex_numberhttp://en.wikipedia.org/wiki/Definable_real_numberhttp://en.wikipedia.org/wiki/Definable_real_numberhttp://en.wikipedia.org/wiki/Exponentiationhttp://en.wikipedia.org/wiki/Pihttp://en.wikipedia.org/wiki/Irrational_numberhttp://en.wikipedia.org/wiki/Real_numberhttp://en.wikipedia.org/wiki/Improper_fractionhttp://en.wikipedia.org/wiki/Proper_fractionhttp://en.wikipedia.org/wiki/Rational_numberhttp://en.wikipedia.org/wiki/Integerhttp://en.wikipedia.org/wiki/Natural_numberhttp://en.wikipedia.org/wiki/Prime_numberhttp://en.wikipedia.org/wiki/Blackboard_bold


7/54

Basic operations

There are several fundamental operations for constructing new sets from given

set.

Unions

The union ofA and B, denoted byAB

Main article: Union (set theory)

Two sets can be "added" together. The union ofA and B, denoted byAB, is

the set of all things which are members of eitherA orB. Examples:

{1, 2} {red, white} = {1, 2, red, white}.

{1, 2, green} {red, white, green} = {1, 2, red, white, green}.

{1, 2} {1, 2} = {1, 2}.

Some basic properties of unions:

AB = BA.

A (BC) = (AB) C.

A (AB).

AA =A.
http://en.wikipedia.org/wiki/Union_%28set_theory%29http://en.wikipedia.org/wiki/File:Venn0111.svghttp://en.wikipedia.org/wiki/Union_%28set_theory%29


8/54

A =A.

AB if and only ifAB = B.

Intersections

A new set can also be constructed by determining which members two sets have

"in common". The intersection ofA and B, denoted by A B, is the set of all

things which are members of bothA and B. IfA B = , thenA and B are said to

be disjoint.

The intersection ofA and B, denotedA B.

Examples:

{1, 2} {red, white} = .

{1, 2, green} {red, white, green} = {green}.

{1, 2} {1, 2} = {1, 2}.

Some basic properties of intersections:

A B = BA.

A (B C) = (A B) C.

A BA.

AA =A.

A = .
http://en.wikipedia.org/wiki/If_and_only_ifhttp://en.wikipedia.org/wiki/File:Venn0001.svghttp://en.wikipedia.org/wiki/File:Venn0001.svghttp://en.wikipedia.org/wiki/If_and_only_if


9/54

AB if and only ifA B =A.

Complements

The relative complement ofB inA

The complement ofA in U

The symmetric difference ofA and B

Main article: Complement (set theory)

Two sets can also be "subtracted". The relative complementofB inA (also called

the set-theoretic difference ofA and B), denoted byA \ B, (orA B) is the set of

all elements which are members ofA but not members ofB. 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.
http://en.wikipedia.org/wiki/If_and_only_ifhttp://en.wikipedia.org/wiki/Complement_%28set_theory%29http://en.wikipedia.org/wiki/File:Venn0110.svghttp://en.wikipedia.org/wiki/File:Venn0100.svghttp://en.wikipedia.org/wiki/File:Venn0110.svghttp://en.wikipedia.org/wiki/File:Venn0100.svghttp://en.wikipedia.org/wiki/Complement_%28set_theory%29http://en.wikipedia.org/wiki/If_and_only_if


10/54

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 complements ofA, and is denoted byA.

Examples:

{1, 2} \ {red, white} = {1, 2}.

{1, 2, green} \ {red, white, green} = {1, 2}.

{1, 2} \ {1, 2} = .

{1, 2, 3, 4} \ {1, 3} = {2, 4}.

IfUis the set of integers, Eis the set of even integers, and O is the

set of odd integers, then E = O.

Some basic properties of complements:

A \ B B \A.

AA = U.

AA = .

(A) =A.

A \A = .

U = and = U.

A \ B =A B.

An extension of the complement is the symmetric difference, defined for setsA,

B as

For example, the symmetric difference of {7,8,9,10} and {9,10,11,12} is the set

{7,8,11,12}.
http://en.wikipedia.org/wiki/Universe_%28mathematics%29http://en.wikipedia.org/wiki/Symmetric_differencehttp://en.wikipedia.org/wiki/Symmetric_differencehttp://en.wikipedia.org/wiki/Universe_%28mathematics%29


11/54

Cartesian product

A new set can be constructed by associating every element of one set with every

element of another set. The Cartesian productof two setsA and B, denoted byA

B is the set of all ordered pairs (a, b) such that a is a member ofA and b is a

member ofB.

Examples:

{1, 2} {red, white} = {(1, red), (1, white), (2, red), (2, white)}.

{1, 2, green} {red, white, green} = {(1, red), (1, white), (1, green),

(2, red), (2, white), (2, green), (green, red), (green, white), (green,

green)}.

{1, 2} {1, 2} = {(1, 1), (1, 2), (2, 1), (2, 2)}.

Some basic properties of Cartesian products:

A = .

A (BC) = (A B) (A C).

(AB) C= (A C) (B C).

LetA and B be finite sets. Then

|A B| = |B A| = |A| |B|.

Applications
http://en.wikipedia.org/wiki/Ordered_pairshttp://en.wikipedia.org/wiki/Ordered_pairs


12/54

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 domainA to a codomain B is a subset of the Cartesian productA

B. Given this concept, we are quick to see that the set Fof 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. 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, y

2

) 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 paradoxIt shows that the "set of all sets which do not containthemselves," i.e. the "set" {x:xis a set andxx} does not exist.

Cantor's paradoxIt 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 axiomatic based on first-order logic, and thus

axiomatic set theorywas born.
http://en.wikipedia.org/wiki/Algebraic_structurehttp://en.wikipedia.org/wiki/Abstract_algebrahttp://en.wikipedia.org/wiki/Group_%28mathematics%29http://en.wikipedia.org/wiki/Field_%28mathematics%29http://en.wikipedia.org/wiki/Ring_%28mathematics%29http://en.wikipedia.org/wiki/Relation_%28mathematics%29http://en.wikipedia.org/wiki/Domain_%28mathematics%29http://en.wikipedia.org/wiki/Codomainhttp://en.wikipedia.org/wiki/Naive_set_theoryhttp://en.wikipedia.org/wiki/Category:Paradoxes_of_naive_set_theoryhttp://en.wikipedia.org/wiki/Russell%27s_paradoxhttp://en.wikipedia.org/wiki/Cantor%27s_paradoxhttp://en.wikipedia.org/wiki/First-order_logichttp://en.wikipedia.org/wiki/Axiomatic_set_theoryhttp://en.wikipedia.org/wiki/Axiomatic_set_theoryhttp://en.wikipedia.org/wiki/Axiomatic_set_theoryhttp://en.wikipedia.org/wiki/First-order_logichttp://en.wikipedia.org/wiki/Cantor%27s_paradoxhttp://en.wikipedia.org/wiki/Russell%27s_paradoxhttp://en.wikipedia.org/wiki/Category:Paradoxes_of_naive_set_theoryhttp://en.wikipedia.org/wiki/Naive_set_theoryhttp://en.wikipedia.org/wiki/Codomainhttp://en.wikipedia.org/wiki/Domain_%28mathematics%29http://en.wikipedia.org/wiki/Relation_%28mathematics%29http://en.wikipedia.org/wiki/Ring_%28mathematics%29http://en.wikipedia.org/wiki/Field_%28mathematics%29http://en.wikipedia.org/wiki/Group_%28mathematics%29http://en.wikipedia.org/wiki/Abstract_algebrahttp://en.wikipedia.org/wiki/Algebraic_structure


13/54

Intensional definition

In logic and mathematics, an intensional definition gives the meaning of a term

by specifying all the properties required to come to that definition, that is, the

necessary and sufficient conditions for belonging to the set being defined.

For example, an intensional definition of "bachelor" is "unmarried man." Being an

unmarried man is an essential property of something referred to as a bachelor. It

is a necessary condition: one cannot be a bachelor without being an unmarried

man. It is also a sufficient condition: any unmarried man is a bachelor.[1]

This is the opposite approach to the extensional definition, which defines by

listing everything that falls under that definition an extensional definition of

"bachelor" would be a listing of all the unmarried men in the world .[1]

As becomes clear, intensional definitions are best used when something has a

clearly-defined set of properties, and it works well for sets that are too large to list

in an extensional definition. It is impossible to give an extensional definition for an

infinite set, but an intensional one can often be stated concisely there is an

infinite number of even numbers, impossible to list, but they can be defined by

saying that even numbers are integermultiples of two.

Definition by genus and difference, in which something is defined by first stating

the broad category it belongs to and then distinguished by specific properties, is

a type of intensional definition. As the name might suggest, this is the type ofdefinition used in Linnaean taxonomy to categorize living things, but is by no

means restricted to biology. Suppose we define a miniskirt as "a skirt with a

hemline above the knee." We've assigned it to a genus, or larger class of items: it

is a type of skirt. Then, we've described the differentia, the specific properties

that make it its own sub-type: it has a hemline above the knee.
http://en.wikipedia.org/wiki/Logichttp://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/Meaning_%28linguistic%29http://en.wikipedia.org/wiki/Definitionhttp://en.wikipedia.org/wiki/Necessary_and_sufficient_conditionshttp://en.wikipedia.org/wiki/Set_%28mathematics%29http://en.wikipedia.org/wiki/Intensional_definition#cite_note-Cook-0http://en.wikipedia.org/wiki/Intensional_definition#cite_note-Cook-0http://en.wikipedia.org/wiki/Intensional_definition#cite_note-Cook-0http://en.wikipedia.org/wiki/Extensional_definitionhttp://en.wikipedia.org/wiki/Extensional_definitionhttp://en.wikipedia.org/wiki/Intensional_definition#cite_note-Cook-0http://en.wikipedia.org/wiki/Intensional_definition#cite_note-Cook-0http://en.wikipedia.org/wiki/Intensional_definition#cite_note-Cook-0http://en.wikipedia.org/wiki/Infinityhttp://en.wikipedia.org/wiki/Even_and_odd_numbershttp://en.wikipedia.org/wiki/Integerhttp://en.wikipedia.org/wiki/Definition_by_genus_and_differencehttp://en.wikipedia.org/wiki/Linnaean_taxonomyhttp://en.wikipedia.org/wiki/Biologyhttp://en.wikipedia.org/wiki/Biologyhttp://en.wikipedia.org/wiki/Linnaean_taxonomyhttp://en.wikipedia.org/wiki/Definition_by_genus_and_differencehttp://en.wikipedia.org/wiki/Integerhttp://en.wikipedia.org/wiki/Even_and_odd_numbershttp://en.wikipedia.org/wiki/Infinityhttp://en.wikipedia.org/wiki/Intensional_definition#cite_note-Cook-0http://en.wikipedia.org/wiki/Extensional_definitionhttp://en.wikipedia.org/wiki/Extensional_definitionhttp://en.wikipedia.org/wiki/Intensional_definition#cite_note-Cook-0http://en.wikipedia.org/wiki/Set_%28mathematics%29http://en.wikipedia.org/wiki/Necessary_and_sufficient_conditionshttp://en.wikipedia.org/wiki/Definitionhttp://en.wikipedia.org/wiki/Meaning_%28linguistic%29http://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/Logic


14/54

Semantics

Semantics (from Greek) is the study of meaning. It typically focuses on the

relation between signifiers, such as words, phrases, signs and symbols, and

what they stand for.

Linguistic semantics is the study of meanings that humans use language to

express. Other forms of semantics include the semantics of programming

languages, formal logics, and semiotics.

The word "semantics" itself denotes a range of ideas, from the popular to the

highly technical. It is often used in ordinary language to denote a problem of

understanding that comes down to word selection or connotation. This problem

of understanding has been the subject of many formal inquiries, over a long

period of time, most notably in the field offormal semantics. In linguistics, it is the

study of interpretation of signs or symbols as used by agents or communities

within particular circumstances and contexts. Within this view, sounds, facial

expressions, body language, proxemics have semantic (meaningful) content, and

each has several branches of study. In written language, such things as

paragraph structure and punctuation have semantic content; in other forms oflanguage, there is other semantic content.

The formal study of semantics intersects with many other fields of inquiry,

including lexicology, syntax, pragmatics, etymology and others, although

semantics is a well-defined field in its own right, often with synthetic properties. In

philosophy of language, semantics and reference are related fields. Further

related fields include philology,communication, and semiotics. The formal study

of semantics is therefore complex.

Semantics contrasts with syntax, the study of the combinatorics of units of a

language (without reference to their meaning), and pragmatics, the study of the

relationships between the symbols of a language, their meaning, and the users of

the language.
http://en.wikipedia.org/wiki/Greek_languagehttp://en.wikipedia.org/wiki/Wordhttp://en.wikipedia.org/wiki/Phrasehttp://en.wikipedia.org/wiki/Signhttp://en.wikipedia.org/wiki/Symbolhttp://en.wikipedia.org/wiki/Connotationhttp://en.wikipedia.org/wiki/Formal_semanticshttp://en.wikipedia.org/wiki/Linguisticshttp://en.wikipedia.org/wiki/Agent_%28grammar%29http://en.wikipedia.org/wiki/Communityhttp://en.wikipedia.org/wiki/Proxemicshttp://en.wikipedia.org/wiki/Lexicologyhttp://en.wikipedia.org/wiki/Syntaxhttp://en.wikipedia.org/wiki/Pragmaticshttp://en.wikipedia.org/wiki/Etymologyhttp://en.wikipedia.org/wiki/Philosophy_of_languagehttp://en.wikipedia.org/wiki/Referencehttp://en.wikipedia.org/wiki/Philologyhttp://en.wikipedia.org/wiki/Communicationhttp://en.wikipedia.org/wiki/Semioticshttp://en.wikipedia.org/wiki/Syntaxhttp://en.wikipedia.org/wiki/Pragmaticshttp://en.wikipedia.org/wiki/Pragmaticshttp://en.wikipedia.org/wiki/Syntaxhttp://en.wikipedia.org/wiki/Semioticshttp://en.wikipedia.org/wiki/Communicationhttp://en.wikipedia.org/wiki/Philologyhttp://en.wikipedia.org/wiki/Referencehttp://en.wikipedia.org/wiki/Philosophy_of_languagehttp://en.wikipedia.org/wiki/Etymologyhttp://en.wikipedia.org/wiki/Pragmaticshttp://en.wikipedia.org/wiki/Syntaxhttp://en.wikipedia.org/wiki/Lexicologyhttp://en.wikipedia.org/wiki/Proxemicshttp://en.wikipedia.org/wiki/Communityhttp://en.wikipedia.org/wiki/Agent_%28grammar%29http://en.wikipedia.org/wiki/Linguisticshttp://en.wikipedia.org/wiki/Formal_semanticshttp://en.wikipedia.org/wiki/Connotationhttp://en.wikipedia.org/wiki/Symbolhttp://en.wikipedia.org/wiki/Signhttp://en.wikipedia.org/wiki/Phrasehttp://en.wikipedia.org/wiki/Wordhttp://en.wikipedia.org/wiki/Greek_language


15/54

Intensional definition also applies to rules or sets of axioms that generate all

members of the set being defined. For example, an intensional definition of

"square number" can be "any number that can be expressed as some integer

multiplied by itself." The rule "take an integer and multiply it by itself"

always generates members of the set of square numbers, no matter which

integer one chooses, and for any square number, there is an integer that was

multiplied by itself to get it.

Similarly, an intensional definition of a game, such as chess, would be the rules

of the game; any game played by those rules must be a game of chess, and any

game properly called a game of chess must have been played by those rules.
http://en.wikipedia.org/wiki/Axiomhttp://en.wikipedia.org/wiki/Square_numberhttp://en.wikipedia.org/wiki/Integerhttp://en.wikipedia.org/wiki/Chesshttp://en.wikipedia.org/wiki/Chesshttp://en.wikipedia.org/wiki/Integerhttp://en.wikipedia.org/wiki/Square_numberhttp://en.wikipedia.org/wiki/Axiom


16/54

Category of sets

In the mathematical field of category theory, the category of sets, denoted as

Set, is the category whose objects are sets. The arrows ormorphisms between

setsA and B are all functions fromA to B. Care must be taken in the definition of

Set to avoid set-theoretic paradoxes.

The category of sets is the most basic and the most commonly used category in

mathematics. Many other categories (such as the category of groups, with group

homomorphisms as arrows) add structure to the objects of this category and/orrestrict the arrows to functions of a particular kind.

Properties of the category of sets

The epimorphisms in Set are the surjective maps, the monomorphisms are the

injective maps, and the isomorphisms are the bijective maps.

The empty set serves as the initial object in Set with empty functions as

morphisms. Every singleton is a terminal object, with the functions mapping all

elements of the source sets to the single target element as morphisms. There are

thus no zero objects in Set.

The category Set is complete and co-complete. The product in this category is

given by the cartesian product of sets. The coproduct is given by the disjoint

union: given sets Ai where i ranges over some index set I, we construct the

coproduct as the union ofAi{i} (the cartesian product with iserves to ensure that

all the components stay disjoint).
http://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/Category_theoryhttp://en.wikipedia.org/wiki/Category_%28mathematics%29http://en.wikipedia.org/wiki/Category_theoryhttp://en.wikipedia.org/wiki/Set_%28mathematics%29http://en.wikipedia.org/wiki/Morphismhttp://en.wikipedia.org/wiki/Function_%28mathematics%29http://en.wikipedia.org/wiki/Paradoxes_of_set_theoryhttp://en.wikipedia.org/wiki/Category_of_groupshttp://en.wikipedia.org/wiki/Group_homomorphismshttp://en.wikipedia.org/wiki/Group_homomorphismshttp://en.wikipedia.org/wiki/Epimorphismhttp://en.wikipedia.org/wiki/Surjectivehttp://en.wikipedia.org/wiki/Monomorphismhttp://en.wikipedia.org/wiki/Injectivehttp://en.wikipedia.org/wiki/Isomorphismhttp://en.wikipedia.org/wiki/Bijectivehttp://en.wikipedia.org/wiki/Empty_sethttp://en.wikipedia.org/wiki/Initial_objecthttp://en.wikipedia.org/wiki/Empty_functionhttp://en.wikipedia.org/wiki/Singleton_%28mathematics%29http://en.wikipedia.org/wiki/Terminal_objecthttp://en.wikipedia.org/wiki/Zero_objecthttp://en.wikipedia.org/wiki/Complete_categoryhttp://en.wikipedia.org/wiki/Product_%28category_theory%29http://en.wikipedia.org/wiki/Cartesian_producthttp://en.wikipedia.org/wiki/Coproduct_%28category_theory%29http://en.wikipedia.org/wiki/Disjoint_unionhttp://en.wikipedia.org/wiki/Disjoint_unionhttp://en.wikipedia.org/wiki/Disjoint_unionhttp://en.wikipedia.org/wiki/Disjoint_unionhttp://en.wikipedia.org/wiki/Coproduct_%28category_theory%29http://en.wikipedia.org/wiki/Cartesian_producthttp://en.wikipedia.org/wiki/Product_%28category_theory%29http://en.wikipedia.org/wiki/Complete_categoryhttp://en.wikipedia.org/wiki/Zero_objecthttp://en.wikipedia.org/wiki/Terminal_objecthttp://en.wikipedia.org/wiki/Singleton_%28mathematics%29http://en.wikipedia.org/wiki/Empty_functionhttp://en.wikipedia.org/wiki/Initial_objecthttp://en.wikipedia.org/wiki/Empty_sethttp://en.wikipedia.org/wiki/Bijectivehttp://en.wikipedia.org/wiki/Isomorphismhttp://en.wikipedia.org/wiki/Injectivehttp://en.wikipedia.org/wiki/Monomorphismhttp://en.wikipedia.org/wiki/Surjectivehttp://en.wikipedia.org/wiki/Epimorphismhttp://en.wikipedia.org/wiki/Group_homomorphismshttp://en.wikipedia.org/wiki/Group_homomorphismshttp://en.wikipedia.org/wiki/Category_of_groupshttp://en.wikipedia.org/wiki/Paradoxes_of_set_theoryhttp://en.wikipedia.org/wiki/Function_%28mathematics%29http://en.wikipedia.org/wiki/Morphismhttp://en.wikipedia.org/wiki/Set_%28mathematics%29http://en.wikipedia.org/wiki/Category_theoryhttp://en.wikipedia.org/wiki/Category_%28mathematics%29http://en.wikipedia.org/wiki/Category_theoryhttp://en.wikipedia.org/wiki/Mathematics


17/54

Set is the prototype of a concrete category; other categories are concrete if they

"resemble" Set in some well-defined way.

Every two-element set serves as a subobject classifierin Set. The power object

of a setA is given by its power set, and the exponential object of the setsA and

B is given by the set of all functions from A to B. Set is thus a topos (and in

particularcartesian closed).

Set is not abelian, additive or preadditive. Its zero morphisms are the empty

functions X.

Every not initial object in Set is injective and (assuming the axiom of choice)

also projective.

Foundations for the category of sets

In ZermeloFraenkel set theory the collection of all sets is not a set; this follows

from the axiom of foundation. One refers to collections that are not sets as proper

classes. One can't handle proper classes as one handles sets; in particular, one

can't write that those proper classes belong to a collection (either a set or a

proper class). This is a problem: it means that the category of sets cannot be

formalized straightforwardly in this setting.

One way to resolve the problem is to work in a system that gives formal status to

proper classes, such as NBG set theory. In this setting, categories formed from

sets are said to be smalland those (like Set) that are formed from proper classes

are said to be large.

Another solution is to assume the existence ofGrothendieck universes. Roughly

speaking, a Grothendieck universe is a set which is itself a model of ZF(C) (for

instance if a set belongs to a universe, its elements and its powerset will belong

to the universe). The existence of Grothendieck universes (other than the empty

set and the set V of all hereditarily finite sets) is not implied by the usual ZF
http://en.wikipedia.org/wiki/Concrete_categoryhttp://en.wikipedia.org/wiki/Subobject_classifierhttp://en.wikipedia.org/wiki/Power_sethttp://en.wikipedia.org/wiki/Exponential_objecthttp://en.wikipedia.org/wiki/Toposhttp://en.wikipedia.org/wiki/Cartesian_closed_categoryhttp://en.wikipedia.org/wiki/Abelian_categoryhttp://en.wikipedia.org/wiki/Additive_categoryhttp://en.wikipedia.org/wiki/Preadditive_categoryhttp://en.wikipedia.org/wiki/Zero_morphismhttp://en.wikipedia.org/wiki/Injective_objecthttp://en.wikipedia.org/wiki/Axiom_of_choicehttp://en.wikipedia.org/wiki/Projective_modulehttp://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theoryhttp://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theoryhttp://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theoryhttp://en.wikipedia.org/wiki/Axiom_of_foundationhttp://en.wikipedia.org/wiki/Proper_classhttp://en.wikipedia.org/wiki/Proper_classhttp://en.wikipedia.org/wiki/NBG_set_theoryhttp://en.wikipedia.org/wiki/Grothendieck_universehttp://en.wikipedia.org/wiki/Hereditarily_finite_sethttp://en.wikipedia.org/wiki/Hereditarily_finite_sethttp://en.wikipedia.org/wiki/Grothendieck_universehttp://en.wikipedia.org/wiki/NBG_set_theoryhttp://en.wikipedia.org/wiki/Proper_classhttp://en.wikipedia.org/wiki/Proper_classhttp://en.wikipedia.org/wiki/Axiom_of_foundationhttp://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theoryhttp://en.wikipedia.org/wiki/Projective_modulehttp://en.wikipedia.org/wiki/Axiom_of_choicehttp://en.wikipedia.org/wiki/Injective_objecthttp://en.wikipedia.org/wiki/Zero_morphismhttp://en.wikipedia.org/wiki/Preadditive_categoryhttp://en.wikipedia.org/wiki/Additive_categoryhttp://en.wikipedia.org/wiki/Abelian_categoryhttp://en.wikipedia.org/wiki/Cartesian_closed_categoryhttp://en.wikipedia.org/wiki/Toposhttp://en.wikipedia.org/wiki/Exponential_objecthttp://en.wikipedia.org/wiki/Power_sethttp://en.wikipedia.org/wiki/Subobject_classifierhttp://en.wikipedia.org/wiki/Concrete_category


18/54

axioms; it is an additional, independent axiom, roughly equivalent to the

existence ofstrongly inaccessible cardinals. Assuming this extra axiom, one can

limit the objects ofSet to the elements of a particular universe. (There is no "set

of all sets" within the model, but one can still reason about the class Uof all inner

sets, i. e., elements ofU.)

In one variation of this scheme, the class of sets is the union of the entire tower

of Grothendieck universes. (This is necessarily a proper class, but each

Grothendieck universe is a set because it is an element of some larger

Grothendieck universe.) However, one does not work directly with the "category

of all sets". Instead, theorems are expressed in terms of the category SetUwhose

objects are the elements of a sufficiently large Grothendieck universe U, and arethen shown not to depend on the particular choice of U. As a foundation for

category theory, this approach is well matched to a system like Tarski-

Grothendieck set theory in which one cannot reason directly about proper

classes; its principal disadvantage is that a theorem can be true of all SetU but

not ofSet.

Various other solutions, and variations on the above, have been proposed.

The same issues arise with other concrete categories, such as the category of

groups or the category of topological spaces.
http://en.wikipedia.org/wiki/Strongly_inaccessible_cardinalhttp://en.wikipedia.org/wiki/Proper_classhttp://en.wikipedia.org/wiki/Category_theoryhttp://en.wikipedia.org/wiki/Tarski-Grothendieck_set_theoryhttp://en.wikipedia.org/wiki/Tarski-Grothendieck_set_theoryhttp://en.wikipedia.org/wiki/Category_of_groupshttp://en.wikipedia.org/wiki/Category_of_groupshttp://en.wikipedia.org/wiki/Category_of_topological_spaceshttp://en.wikipedia.org/wiki/Category_of_topological_spaceshttp://en.wikipedia.org/wiki/Category_of_groupshttp://en.wikipedia.org/wiki/Category_of_groupshttp://en.wikipedia.org/wiki/Tarski-Grothendieck_set_theoryhttp://en.wikipedia.org/wiki/Tarski-Grothendieck_set_theoryhttp://en.wikipedia.org/wiki/Category_theoryhttp://en.wikipedia.org/wiki/Proper_classhttp://en.wikipedia.org/wiki/Strongly_inaccessible_cardinal


19/54

Set theory

Set theory is the branch ofmathematics that studies sets, which are collections

of objects. Although any type of object can be collected into a set, set theory is

applied most often to objects that are relevant to mathematics.

The modern study of set theory was initiated by Georg Cantor and Richard

Dedekind in the 1870s. After the discovery of paradoxes in naive set theory,

numerous axiom systems were proposed in the early twentieth century, of which

the ZermeloFraenkel axioms, with the axiom of choice, are the best-known.

The language of set theory is used in the definitions of nearly all mathematical

objects, such as functions, and concepts of set theory are integrated throughout

the mathematics curriculum. Elementary facts about sets and set membership

can be introduced in primary school, along with Venn and Euler diagrams, to

study collections of commonplace physical objects. Elementary operations such

as set union and intersection can be studied in this context. More advanced

concepts such as cardinality are a standard part of the undergraduate

mathematics curriculum.

Set theory is commonly employed as a foundational system for mathematics,

particularly in the form ofZermeloFraenkel set theory with the axiom of choice.

Beyond its foundational role, set theory is a branch of mathematics in its own

right, with an active research community. Contemporary research into set theory

includes a diverse collection of topics, ranging from the structure of the real

numberline to the study of the consistency oflarge cardinals.
http://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/Set_%28mathematics%29http://en.wikipedia.org/wiki/Georg_Cantorhttp://en.wikipedia.org/wiki/Richard_Dedekindhttp://en.wikipedia.org/wiki/Richard_Dedekindhttp://en.wikipedia.org/wiki/Paradoxes_of_set_theoryhttp://en.wikipedia.org/wiki/Naive_set_theoryhttp://en.wikipedia.org/wiki/Axiomatic_systemhttp://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theoryhttp://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theoryhttp://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theoryhttp://en.wikipedia.org/wiki/Axiom_of_choicehttp://en.wikipedia.org/wiki/Function_%28mathematics%29http://en.wikipedia.org/wiki/Venn_diagramhttp://en.wikipedia.org/wiki/Euler_diagramshttp://en.wikipedia.org/wiki/Cardinalityhttp://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theoryhttp://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theoryhttp://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theoryhttp://en.wikipedia.org/wiki/Axiom_of_choicehttp://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/Real_numberhttp://en.wikipedia.org/wiki/Real_numberhttp://en.wikipedia.org/wiki/Consistencyhttp://en.wikipedia.org/wiki/Large_cardinalhttp://en.wikipedia.org/wiki/Large_cardinalhttp://en.wikipedia.org/wiki/Consistencyhttp://en.wikipedia.org/wiki/Real_numberhttp://en.wikipedia.org/wiki/Real_numberhttp://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/Axiom_of_choicehttp://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theoryhttp://en.wikipedia.org/wiki/Cardinalityhttp://en.wikipedia.org/wiki/Euler_diagramshttp://en.wikipedia.org/wiki/Venn_diagramhttp://en.wikipedia.org/wiki/Function_%28mathematics%29http://en.wikipedia.org/wiki/Axiom_of_choicehttp://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theoryhttp://en.wikipedia.org/wiki/Axiomatic_systemhttp://en.wikipedia.org/wiki/Naive_set_theoryhttp://en.wikipedia.org/wiki/Paradoxes_of_set_theoryhttp://en.wikipedia.org/wiki/Richard_Dedekindhttp://en.wikipedia.org/wiki/Richard_Dedekindhttp://en.wikipedia.org/wiki/Georg_Cantorhttp://en.wikipedia.org/wiki/Set_%28mathematics%29http://en.wikipedia.org/wiki/Mathematics


20/54

Class (set theory)

In set theory and its applications throughout mathematics, a class is a collection

ofsets (or sometimes other mathematical objects) which can be unambiguously

defined by a property that all its members share. The precise definition of "class"

depends on foundational context. In work on ZF set theory, the notion of class is

informal, whereas other set theories, such as NBG set theory, axiomatize the

notion of "class".

Every set is a class, no matter which foundation is chosen. A class that is not a

set (informally in ZermeloFraenkel) is called a proper class, and a class that is

a set is sometimes called a small class. For instance, the class of all ordinal

number, and the class of all set are proper classes in many formal systems.

Outside set theory, the word "class" is sometimes used synonymously with "set".

This usage dates from a historical period where classes and sets were not

distinguished as they are in modern set-theoretic terminology. Many discussions

of "classes" in the 19th century and earlier are really referring to sets, or perhaps

to a more ambiguous concept.

Examples

The collection of all algebraic objects of a given type will usually be a proper

class. Examples include the class of all groups, the class of all vector spaces,

and many others. In category theory, a category whose collection of objects

forms a proper class (or whose collection of morphisms forms a proper class) is

called a large category.

The surreal numbers are a proper class of objects that has the properties of a

field.
http://en.wikipedia.org/wiki/Set_theoryhttp://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/Set_%28mathematics%29http://en.wikipedia.org/wiki/ZF_set_theoryhttp://en.wikipedia.org/wiki/NBG_set_theoryhttp://en.wikipedia.org/wiki/Group_%28mathematics%29http://en.wikipedia.org/wiki/Vector_spacehttp://en.wikipedia.org/wiki/Category_theoryhttp://en.wikipedia.org/wiki/Large_categoryhttp://en.wikipedia.org/wiki/Surreal_numberhttp://en.wikipedia.org/wiki/Field_%28mathematics%29http://en.wikipedia.org/wiki/Field_%28mathematics%29http://en.wikipedia.org/wiki/Surreal_numberhttp://en.wikipedia.org/wiki/Large_categoryhttp://en.wikipedia.org/wiki/Category_theoryhttp://en.wikipedia.org/wiki/Vector_spacehttp://en.wikipedia.org/wiki/Group_%28mathematics%29http://en.wikipedia.org/wiki/NBG_set_theoryhttp://en.wikipedia.org/wiki/ZF_set_theoryhttp://en.wikipedia.org/wiki/Set_%28mathematics%29http://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/Set_theory


21/54

Within set theory, many collections of sets turn out to be proper classes.

Examples include the class of all sets, the class of all ordinal numbers, and the

class of all cardinal numbers.

One way to prove that a class is proper is to place it in bijection with the class of

all ordinal numbers. This method is used, for example, in the proof that there is

no free complete lattice.

Paradoxes

The paradoxes of naive set theory can be explained in terms of the inconsistent

assumption that "all classes are sets". With a rigorous foundation, these

paradoxes instead suggest proofs that certain classes are proper. For example,

Russell's paradox suggests a proof that the class of all sets which do not contain

themselves is proper, and the Burali-Forti paradox suggests that the class of all

ordinal numbers is proper.
http://en.wikipedia.org/wiki/Free_lattice#The_complete_free_latticehttp://en.wikipedia.org/wiki/Naive_set_theory#Paradoxeshttp://en.wikipedia.org/wiki/Proof_%28mathematics%29http://en.wikipedia.org/wiki/Russell%27s_paradoxhttp://en.wikipedia.org/wiki/Burali-Forti_paradoxhttp://en.wikipedia.org/wiki/Ordinal_numbershttp://en.wikipedia.org/wiki/Ordinal_numbershttp://en.wikipedia.org/wiki/Burali-Forti_paradoxhttp://en.wikipedia.org/wiki/Russell%27s_paradoxhttp://en.wikipedia.org/wiki/Proof_%28mathematics%29http://en.wikipedia.org/wiki/Naive_set_theory#Paradoxeshttp://en.wikipedia.org/wiki/Free_lattice#The_complete_free_lattice


22/54

Classes in formal set theories

ZF set theory does not formalize the notion of classes. They can instead be

described in the metalanguage, as equivalence classes of logical formulas. For

example, if is a structure interpreting ZF, then the metalanguage expression

is interpreted in by the collection of all the elements from the

domain of ; that is, all the sets in . So we can identify the "class of all sets"

with the predicatex=xor any equivalent predicate.

Because classes do not have any formal status in the theory of ZF, the axioms of

ZF do not immediately apply to classes. However, if an inaccessible cardinal is

assumed, then the sets of smaller rank form a model of ZF (a Grothendieck

universe), and its subsets can be thought of as "classes".

Another approach is taken by the von NeumannBernaysGdel axioms (NBG);

classes are the basic objects in this theory, and a set is then defined to be a

class that is an element of some other class. However, the set existence axioms

of NBG are restricted so that they only quantify over sets, rather than over all

classes. This causes NBG to be a conservative extension of ZF.

MorseKelley set theory admits proper classes as basic objects, like NBG, but

also allows quantification over all proper classes in its set existence axioms. This

causes MK to be strictly stronger than both NBG and ZF.

In other set theories, such as New Foundations or the theory of semisets, the

concept of "proper class" still makes sense (not all classes are sets) but the

criterion of sethood is not closed under subsets. For example, any set theory with

a universal set has proper classes which are subclasses of sets.
http://en.wikipedia.org/wiki/ZF_set_theoryhttp://en.wikipedia.org/wiki/Metalanguagehttp://en.wikipedia.org/wiki/Structure_%28mathematical_logic%29http://en.wikipedia.org/wiki/Inaccessible_cardinalhttp://en.wikipedia.org/wiki/Grothendieck_universehttp://en.wikipedia.org/wiki/Grothendieck_universehttp://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_axiomshttp://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_axiomshttp://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_axiomshttp://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_axiomshttp://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_axiomshttp://en.wikipedia.org/wiki/Conservative_extensionhttp://en.wikipedia.org/wiki/Morse%E2%80%93Kelley_set_theoryhttp://en.wikipedia.org/wiki/Morse%E2%80%93Kelley_set_theoryhttp://en.wikipedia.org/wiki/Morse%E2%80%93Kelley_set_theoryhttp://en.wikipedia.org/wiki/New_Foundationshttp://en.wikipedia.org/wiki/Semisethttp://en.wikipedia.org/wiki/Semisethttp://en.wikipedia.org/wiki/New_Foundationshttp://en.wikipedia.org/wiki/Morse%E2%80%93Kelley_set_theoryhttp://en.wikipedia.org/wiki/Conservative_extensionhttp://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_axiomshttp://en.wikipedia.org/wiki/Grothendieck_universehttp://en.wikipedia.org/wiki/Grothendieck_universehttp://en.wikipedia.org/wiki/Inaccessible_cardinalhttp://en.wikipedia.org/wiki/Structure_%28mathematical_logic%29http://en.wikipedia.org/wiki/Metalanguagehttp://en.wikipedia.org/wiki/ZF_set_theory


23/54

Family of sets

In set theory and related branches ofmathematics, a collection Fofsubsets of a

given set S is called a family of subsets ofS, or a family of sets overS. More

generally, a collection of any sets whatsoever is called a family of sets.

Examples

The power set P(S) is a family of sets overS.

Thek-subsets S(k) of ann-set S form a family of sets.

The class Ord of all ordinal numbers is a large family of sets; that is, it is

not itself a set but instead a proper class.

Properties

Any family of subsets ofS is itself a subset of the power set P(S).

Any family of sets whatsoever is a subclass of the proper class V of allsets (the universe).

Related concepts

Certain types of objects from other areas of mathematics are equivalent to

families of sets, in that they can be described purely as a collection of sets of

objects of some type:

A hypergraph, also called a set system, is formed by a set of vertices

together with another set of hyperedges, each of which may be an

arbitrary set. The hyperedges of a hypergraph form a family of sets, and
http://en.wikipedia.org/wiki/Set_theoryhttp://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/Subsethttp://en.wikipedia.org/wiki/Set_%28mathematics%29http://en.wikipedia.org/wiki/Power_sethttp://en.wikipedia.org/wiki/N-sethttp://en.wikipedia.org/wiki/N-sethttp://en.wikipedia.org/wiki/N-sethttp://en.wikipedia.org/wiki/N-sethttp://en.wikipedia.org/wiki/N-sethttp://en.wikipedia.org/wiki/N-sethttp://en.wikipedia.org/wiki/Ordinal_numberhttp://en.wikipedia.org/wiki/Proper_classhttp://en.wikipedia.org/wiki/Subclass_%28sets%29http://en.wikipedia.org/wiki/Universe_%28math%29http://en.wikipedia.org/wiki/Hypergraphhttp://en.wikipedia.org/wiki/Vertex_%28graph_theory%29http://en.wikipedia.org/wiki/Vertex_%28graph_theory%29http://en.wikipedia.org/wiki/Hypergraphhttp://en.wikipedia.org/wiki/Universe_%28math%29http://en.wikipedia.org/wiki/Subclass_%28sets%29http://en.wikipedia.org/wiki/Proper_classhttp://en.wikipedia.org/wiki/Ordinal_numberhttp://en.wikipedia.org/wiki/N-sethttp://en.wikipedia.org/wiki/N-sethttp://en.wikipedia.org/wiki/Power_sethttp://en.wikipedia.org/wiki/Set_%28mathematics%29http://en.wikipedia.org/wiki/Subsethttp://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/Set_theory


24/54

any family of sets can be interpreted as a hypergraph that has the union of

the sets as its vertices.

An abstract simplicial complex is a combinatorial abstraction of the notion

of a simplicial complex, a shape formed by unions of line segments,

triangles, tetrahedra, and higher dimensional simplices, joined face to

face. In an abstract simplicial complex, each simplex is represented simply

as the set of its vertices. Any family of finite sets in which the subsets of

any set in the family also belong to the family forms an abstract simplicial

complex.

An incidence structure consists of a set ofpoints, a set of lines, and an

(arbitrary) binary relation specifying which points belong to which lines. If

no two lines contain the same set of points, an incidence structure can be

specified by a family of sets, the sets of points belonging to each line, and

any family of sets can be interpreted as an incidence structure in this way.
http://en.wikipedia.org/wiki/Abstract_simplicial_complexhttp://en.wikipedia.org/wiki/Simplicial_complexhttp://en.wikipedia.org/wiki/Simplexhttp://en.wikipedia.org/wiki/Incidence_structurehttp://en.wikipedia.org/wiki/Binary_relationhttp://en.wikipedia.org/wiki/Binary_relationhttp://en.wikipedia.org/wiki/Incidence_structurehttp://en.wikipedia.org/wiki/Simplexhttp://en.wikipedia.org/wiki/Simplicial_complexhttp://en.wikipedia.org/wiki/Abstract_simplicial_complex


25/54

Power set

The elements of the power set of the set {x, y, z} ordered in respect to inclusion.

In mathematics, given a set S, the power set (orpower set) ofS, written ,

P(S), (S) or2S, is the set of all subsets ofS, including the empty set and S

itself. In axiomatic set theory (as developed e.g. in the ZFC axioms), the

existence of the power set of any set is postulated by the axiom of power set.

Any subset Fof is called afamily of setsoverS. Example:

IfS is the set {x, y, z}, then the subsets ofS are:

{} (also denoted , the empty set)

{x}

{y}

{z}

{x, y}

{x, z}

{y, z}

{x, y, z}
http://en.wikipedia.org/wiki/Order_theoryhttp://en.wikipedia.org/wiki/Inclusion_%28set_theory%29http://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/Set_%28mathematics%29http://en.wikipedia.org/wiki/Weierstrass_phttp://en.wikipedia.org/wiki/Weierstrass_phttp://en.wikipedia.org/wiki/Power_set#Representing_subsets_as_functionshttp://en.wikipedia.org/wiki/Power_set#Representing_subsets_as_functionshttp://en.wikipedia.org/wiki/Power_set#Representing_subsets_as_functionshttp://en.wikipedia.org/wiki/Subsethttp://en.wikipedia.org/wiki/Axiomatic_set_theoryhttp://en.wikipedia.org/wiki/ZFChttp://en.wikipedia.org/wiki/Axiom_of_power_sethttp://en.wikipedia.org/wiki/Subsethttp://en.wikipedia.org/wiki/Family_of_setshttp://en.wikipedia.org/wiki/Family_of_setshttp://en.wikipedia.org/wiki/Family_of_setshttp://en.wikipedia.org/wiki/Empty_sethttp://en.wikipedia.org/wiki/File:Hasse_diagram_of_powerset_of_3.svghttp://en.wikipedia.org/wiki/File:Hasse_diagram_of_powerset_of_3.svghttp://en.wikipedia.org/wiki/File:Hasse_diagram_of_powerset_of_3.svghttp://en.wikipedia.org/wiki/File:Hasse_diagram_of_powerset_of_3.svghttp://en.wikipedia.org/wiki/Empty_sethttp://en.wikipedia.org/wiki/Family_of_setshttp://en.wikipedia.org/wiki/Subsethttp://en.wikipedia.org/wiki/Axiom_of_power_sethttp://en.wikipedia.org/wiki/ZFChttp://en.wikipedia.org/wiki/Axiomatic_set_theoryhttp://en.wikipedia.org/wiki/Subsethttp://en.wikipedia.org/wiki/Power_set#Representing_subsets_as_functionshttp://en.wikipedia.org/wiki/Weierstrass_phttp://en.wikipedia.org/wiki/Set_%28mathematics%29http://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/Inclusion_%28set_theory%29http://en.wikipedia.org/wiki/Order_theory


26/54

hence the power set ofS is

Properties

If S is a finite set with |S| = n elements, then the power set of S contains

elements.

Cantor's diagonal argument shows that the power set of a set (whether infinite or

not) always has strictly highercardinality than the set itself (informally the power

set must be larger than the original set). In particular, Cantor's theorem shows

that the power set of a countably infinite set is uncountably infinite. For example,

the power set of the set of natural numbers can be put in a one-to-one

correspondence with the set ofreal numbers (see cardinality of the continuum).

The power set of a set S, together with the operations ofunion, intersection and

complement can be viewed as the prototypical example of a Boolean algebra. In

fact, one can show that any finite Boolean algebra is isomorphic to the Boolean

algebra of the power set of a finite set. For infinite Boolean algebras this is no

longer true, but every infinite Boolean algebra is a subalgebra of a power set

Boolean algebra (see Stone's representation theorem).

The power set of a set S forms an Abelian group when considered with the

operation of symmetric difference (with the empty set as its unit and each set

being its own inverse) and a commutative monoid when considered with the

operation of intersection. It can hence be shown (by proving the distributive laws)

that the power set considered together with both of these operations forms a

commutative ring.
http://en.wikipedia.org/wiki/Cantor%27s_diagonal_argumenthttp://en.wikipedia.org/wiki/Cardinalityhttp://en.wikipedia.org/wiki/Cantor%27s_theoremhttp://en.wikipedia.org/wiki/Countable_sethttp://en.wikipedia.org/wiki/Uncountablehttp://en.wikipedia.org/wiki/Natural_numberhttp://en.wikipedia.org/wiki/Bijectionhttp://en.wikipedia.org/wiki/Bijectionhttp://en.wikipedia.org/wiki/Real_numberhttp://en.wikipedia.org/wiki/Cardinality_of_the_continuumhttp://en.wikipedia.org/wiki/Union_%28set_theory%29http://en.wikipedia.org/wiki/Intersection_%28set_theory%29http://en.wikipedia.org/wiki/Complement_%28set_theory%29http://en.wikipedia.org/wiki/Boolean_algebra_%28structure%29http://en.wikipedia.org/wiki/Stone%27s_representation_theoremhttp://en.wikipedia.org/wiki/Abelian_grouphttp://en.wikipedia.org/wiki/Symmetric_differencehttp://en.wikipedia.org/wiki/Commutativehttp://en.wikipedia.org/wiki/Monoidhttp://en.wikipedia.org/wiki/Ring_%28mathematics%29http://en.wikipedia.org/wiki/Ring_%28mathematics%29http://en.wikipedia.org/wiki/Monoidhttp://en.wikipedia.org/wiki/Commutativehttp://en.wikipedia.org/wiki/Symmetric_differencehttp://en.wikipedia.org/wiki/Abelian_grouphttp://en.wikipedia.org/wiki/Stone%27s_representation_theoremhttp://en.wikipedia.org/wiki/Boolean_algebra_%28structure%29http://en.wikipedia.org/wiki/Complement_%28set_theory%29http://en.wikipedia.org/wiki/Intersection_%28set_theory%29http://en.wikipedia.org/wiki/Union_%28set_theory%29http://en.wikipedia.org/wiki/Cardinality_of_the_continuumhttp://en.wikipedia.org/wiki/Real_numberhttp://en.wikipedia.org/wiki/Bijectionhttp://en.wikipedia.org/wiki/Bijectionhttp://en.wikipedia.org/wiki/Natural_numberhttp://en.wikipedia.org/wiki/Uncountablehttp://en.wikipedia.org/wiki/Countable_sethttp://en.wikipedia.org/wiki/Cantor%27s_theoremhttp://en.wikipedia.org/wiki/Cardinalityhttp://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument


27/54

Representing subsets as functions

In set theory, XY is the set of all functions from Y toX. As 2 can be defined as

{0,1} (see natural number), 2S

(i.e., {0,1}S) is the set of all functions from S to

{0,1}. By identifying a function in 2S

with the corresponding preimage of 1, we see

that there is a bijection between 2S

and , where each function is the

characteristic function of the subset in with which it is identified. Hence 2S

and could be considered identical set-theoretically. (Thus there are two

distinct notational motivations for denoting the power set by 2S: the fact that this

function-representation of subsets makes it a special case of theXYnotation and

the property, mentioned above, that |2S| = 2|S|.)

We can apply this notion to the example above to see the isomorphism with the

binary numbers from 0 to 2n-1 with n being the number of elements in the set. In

S, a 1 in the position corresponding to the location in the set indicates the

presence of the element. So {x, y} = 110

For the whole power set ofS we get:

{ } = 000 (Binary) = 0 (Decimal)

{x} = 100 = 4

{y} = 010 = 2

{z} = 001 = 1

{x, y} = 110 = 6

{x, z} = 101 = 5

{y, z} = 011 = 3

{x, y, z} = 111 = 7
http://en.wikipedia.org/wiki/Function_%28mathematics%29#Function_spaceshttp://en.wikipedia.org/wiki/Function_%28mathematics%29#Function_spaceshttp://en.wikipedia.org/wiki/Function_%28mathematics%29#Function_spaceshttp://en.wikipedia.org/wiki/Function_%28mathematics%29http://en.wikipedia.org/wiki/Natural_number#A_standard_constructionhttp://en.wikipedia.org/wiki/Function_%28mathematics%29http://en.wikipedia.org/wiki/Preimagehttp://en.wikipedia.org/wiki/Bijectionhttp://en.wikipedia.org/wiki/Indicator_functionhttp://en.wikipedia.org/wiki/Set_notation#Metaphor_in_denoting_setshttp://en.wikipedia.org/wiki/Power_set#Propertieshttp://en.wikipedia.org/wiki/Power_set#Propertieshttp://en.wikipedia.org/wiki/Set_notation#Metaphor_in_denoting_setshttp://en.wikipedia.org/wiki/Indicator_functionhttp://en.wikipedia.org/wiki/Bijectionhttp://en.wikipedia.org/wiki/Preimagehttp://en.wikipedia.org/wiki/Function_%28mathematics%29http://en.wikipedia.org/wiki/Natural_number#A_standard_constructionhttp://en.wikipedia.org/wiki/Function_%28mathematics%29http://en.wikipedia.org/wiki/Function_%28mathematics%29#Function_spaces


28/54

Relation to binomial theorem

The power set is closely related to the binomial theorem. The number of sets with

kelements in the power set of a set with n elements will be a combination C(n,k),

also called a binomial coefficient.

For example the power set of a set with three elements, has:

C(3,0) = 1 set with 0 elements

C(3,1) = 3 sets with 1 element

C(3,2) = 3 sets with 2 elements

C(3,3) = 1 set with 3 elements.

Algorithms

If is a finite set, there is a recursive algorithm to calculate .

Define the operation

In English, return the set with the element added to each set in .

If ,then is returned.

Otherwise:

Let be any single element of .

Let , where ' ' denotes the relative

complement of in .

And the result: is returned.

In other words, the power set of the empty set is the set containing the empty set

and the power set of any other set is all the subsets of the set containing some
http://en.wikipedia.org/wiki/Binomial_theoremhttp://en.wikipedia.org/wiki/Combinationhttp://en.wikipedia.org/wiki/Binomial_coefficienthttp://en.wikipedia.org/wiki/Finite_sethttp://en.wikipedia.org/wiki/Complement_%28set_theory%29http://en.wikipedia.org/wiki/Complement_%28set_theory%29http://en.wikipedia.org/wiki/Complement_%28set_theory%29http://en.wikipedia.org/wiki/Complement_%28set_theory%29http://en.wikipedia.org/wiki/Finite_sethttp://en.wikipedia.org/wiki/Binomial_coefficienthttp://en.wikipedia.org/wiki/Combinationhttp://en.wikipedia.org/wiki/Binomial_theorem


29/54

specific element and all the subsets of the set not containing that specific

element.

There are other more efficient ways to calculate the power set. For example, use

a list of the n elements of S to fix a mapping from the bit positions of n-bit

numbers to those elements; then with a simple loop run through all the 2n

numbers representable with n bits, and for each contribute the subset of S

corresponding to the bits that are set (to 1) in the number. When n exceeds the

word-length of the computer, typically 64 in modern CPUs but greater in modern

GPUs, the representation is naturally extended by using an array of words

instead of a single word.

Subsets of limited cardinality

The set of subsets of S of cardinality less than is denoted by or

Similarly, the set of non-empty subsets of S might be denoted by

Topologization of power set

Since any family of functions XY from Y to Xmight be topologized establishing

the so-called function space, the same can be done with the power set 2S

identified as {0,1}S. This particular type of function space is often called a

hyperspace and the topology on the power set is referred to as hypertopology.

Power object

A set can be regarded as an algebra having no nontrivial operations or defining

equations. From this perspective the idea of the power set ofX as the set of

subsets ofXgeneralizes naturally to the subalgebras of an algebraic structure or

algebra.
http://en.wikipedia.org/wiki/Bithttp://en.wikipedia.org/wiki/CPUhttp://en.wikipedia.org/wiki/GPUhttp://en.wikipedia.org/wiki/Cardinalityhttp://en.wikipedia.org/wiki/Function_spacehttp://en.wikipedia.org/wiki/Algebraic_structurehttp://en.wikipedia.org/wiki/Algebraic_structurehttp://en.wikipedia.org/wiki/Function_spacehttp://en.wikipedia.org/wiki/Cardinalityhttp://en.wikipedia.org/wiki/GPUhttp://en.wikipedia.org/wiki/CPUhttp://en.wikipedia.org/wiki/Bit


30/54

Now the power set of a set, when ordered by inclusion, is always a complete

atomic Boolean algebra, and every complete atomic Boolean algebra arises as

the lattice of all subsets of some set. The generalization to arbitrary algebras is

that the set of subalgebras of an algebra, again ordered by inclusion, is always

an algebraic lattice, and every algebraic lattice arises as the lattice of

subalgebras of some algebra. So in that regard subalgebras behave analogously

to subsets.

However there are two important properties of subsets that do not carry over to

subalgebras in general. Firstly, although the subsets of a set form a set (as well

as a lattice), in some classes it may not be possible to organize the subalgebras

of an algebra as itself an algebra in that class, although they can always beorganized as a lattice. Secondly, whereas the subsets of a set are in bijection

with the functions from that set to the set {0,1} = 2, there is no guarantee that a

class of algebras contains an algebra that can play the role of 2 in this way.

Certain classes of algebras do enjoy both these properties. The first property is

more common, the case of having both is relatively rare. One class that does

have both is that of multigraphs. Given two multigraphs G and H, a

homomorphism h: G H consists of two functions, one mapping vertices tovertices and the other mapping edges to edges. The set HG of homomorphisms

from G to Hcan then be organized as the graph whose vertices and edges are

respectively the vertex and edge functions appearing in that set. Furthermore the

subgraphs of a multigraph G are in bijection with the graph homomorphisms from

G to the multigraph definable as the complete directed graph on two vertices

(hence four edges, namely two self-loops and two more edges forming a cycle)

augmented with a fifth edge, namely a second self-loop at one of the vertices.

We can therefore organize the subgraphs ofGas the multigraph G, called the

power object ofG.

What is special about a multigraph as an algebra is that its operations are unary.

A multigraph has two sorts of elements forming a set V of vertices and E of
http://en.wikipedia.org/wiki/Lattice_%28mathematics%29http://en.wikipedia.org/wiki/Algebraic_latticehttp://en.wikipedia.org/wiki/Multigraphhttp://en.wikipedia.org/wiki/Complete_graphhttp://en.wikipedia.org/wiki/Complete_graphhttp://en.wikipedia.org/wiki/Multigraphhttp://en.wikipedia.org/wiki/Algebraic_latticehttp://en.wikipedia.org/wiki/Lattice_%28mathematics%29


31/54

edges, and has two unary operations s,t: E V giving the source (start) and

target (end) vertices of each edge. An algebra all of whose operations are unary

is called a presheaf. Every class of presheaves contains a presheaf that p lays

the role for subalgebras that 2 plays for subsets. Such a class is a special case

of the more general notion of elementary topos as a category that is closed (and

moreover cartesian closed) and has an object , called a subobject classifier.

Although the term "power object" is sometimes used synonymously with

exponential object YX

, in topos theory Yis required to be

Fuzzy set

Fuzzy sets are sets whose elements have degrees of membership. Fuzzy setswere introduced by Lotfi A. Zadeh (1965) as an extension of the classical notion

ofset. In classical set theory, the membership of elements in a set is assessed in

binary terms according to a bivalent condition an element either belongs or

does not belong to the set. By contrast, fuzzy set theory permits the gradual

assessment of the membership of elements in a set; this is described with the aid

of a membership function valued in the real unit interval [0, 1]. Fuzzy sets

generalize classical sets, since the indicator functions of classical sets are

special cases of the membership functions of fuzzy sets, if the latter only take

values 0 or 1. Classical bivalent sets are in fuzzy set theory usually called crisp

sets. The fuzzy set theory can be used in a wide range of domains in which

information is incomplete or imprecise, such as bioinformatics.

Definition

A fuzzy set is a pair (A,m) whereA is a set and .
http://en.wikipedia.org/wiki/Presheafhttp://en.wikipedia.org/wiki/Presheafhttp://en.wikipedia.org/wiki/Toposhttp://en.wikipedia.org/wiki/Category_%28mathematics%29http://en.wikipedia.org/wiki/Closed_categoryhttp://en.wikipedia.org/wiki/Cartesian_closed_categoryhttp://en.wikipedia.org/wiki/Cartesian_closed_categoryhttp://en.wikipedia.org/wiki/Subobject_classifierhttp://en.wikipedia.org/wiki/Exponential_objecthttp://en.wikipedia.org/wiki/Lotfi_Asker_Zadehhttp://en.wikipedia.org/wiki/Set_%28mathematics%29http://en.wikipedia.org/wiki/Set_theoryhttp://en.wikipedia.org/wiki/Principle_of_bivalencehttp://en.wikipedia.org/wiki/Membership_function_%28mathematics%29http://en.wikipedia.org/wiki/Indicator_functionhttp://en.wikipedia.org/wiki/Indicator_functionhttp://en.wikipedia.org/wiki/Membership_function_%28mathematics%29http://en.wikipedia.org/wiki/Principle_of_bivalencehttp://en.wikipedia.org/wiki/Set_theoryhttp://en.wikipedia.org/wiki/Set_%28mathematics%29http://en.wikipedia.org/wiki/Lotfi_Asker_Zadehhttp://en.wikipedia.org/wiki/Exponential_objecthttp://en.wikipedia.org/wiki/Subobject_classifierhttp://en.wikipedia.org/wiki/Cartesian_closed_categoryhttp://en.wikipedia.org/wiki/Closed_categoryhttp://en.wikipedia.org/wiki/Category_%28mathematics%29http://en.wikipedia.org/wiki/Toposhttp://en.wikipedia.org/wiki/Presheaf


32/54

For each , m(x) is called the grade of membership ofx in (A,m). For a

finite setA = {x1,...,xn}, the fuzzy set (A,m) is often denoted by {m(x1) /x1,...,m(xn)

/xn}.

Let . Thenxis called not included in the fuzzy set (A,m) ifm(x) = 0,xis

called fully included ifm(x) = 1, andx is called fuzzy member if 0 < m(x) < 1.

The set is called the support of (A,m) and the set

is called its kernel.

Sometimes, more general variants of the notion of fuzzy set are used, with

membership functions taking values in a (fixed or variable) algebra orstructure L

of a given kind; usually it is required that L be at least a poset or lattice. The

usual membership functions with values in [0, 1] are then called [0, 1]-valued

membership functions.

Fuzzy logic

As an extension of the case ofmulti-valued logic, valuations ( ) of

propositional variables (Vo) into a set of membership degrees (W) can be thought

of as membership functions mapping predicates into fuzzy sets (or more formally,

into an ordered set of fuzzy pairs, called a fuzzy relation). With these valuations,

many-valued logic can be extended to allow for fuzzy premises from which

graded conclusions may be drawn.

This extension is sometimes called "fuzzy logic in the narrow sense" as opposed

to "fuzzy logic in the wider sense," which originated in the engineering fields of

automated control and knowledge engineering, and which encompasses many

topics involving fuzzy sets and "approximated reasoning. Industrial applications

of fuzzy sets in the context of "fuzzy logic in the wider sense" can be found at

fuzzy logic.
http://en.wikipedia.org/wiki/Algebraic_structurehttp://en.wikipedia.org/wiki/Structure_%28mathematical_logic%29http://en.wikipedia.org/wiki/Posethttp://en.wikipedia.org/wiki/Lattice_%28order%29http://en.wikipedia.org/wiki/Multi-valued_logichttp://en.wikipedia.org/wiki/Propositional_variablehttp://en.wikipedia.org/wiki/Membership_function_%28mathematics%29http://en.wikipedia.org/wiki/First-order_logichttp://en.wikipedia.org/wiki/Premiseshttp://en.wikipedia.org/wiki/Engineeringhttp://en.wikipedia.org/wiki/Automationhttp://en.wikipedia.org/wiki/Knowledge_engineeringhttp://en.wikipedia.org/wiki/Fuzzy_logichttp://en.wikipedia.org/wiki/Fuzzy_logichttp://en.wikipedia.org/wiki/Knowledge_engineeringhttp://en.wikipedia.org/wiki/Automationhttp://en.wikipedia.org/wiki/Engineeringhttp://en.wikipedia.org/wiki/Premiseshttp://en.wikipedia.org/wiki/First-order_logichttp://en.wikipedia.org/wiki/Membership_function_%28mathematics%29http://en.wikipedia.org/wiki/Propositional_variablehttp://en.wikipedia.org/wiki/Multi-valued_logichttp://en.wikipedia.org/wiki/Lattice_%28order%29http://en.wikipedia.org/wiki/Posethttp://en.wikipedia.org/wiki/Structure_%28mathematical_logic%29http://en.wikipedia.org/wiki/Algebraic_structure


33/54

Fuzzy number

A fuzzy number is a convex,normalized fuzzy set whose membership

function is at least segmentally continuous and has the functional value A(x) = 1

at precisely one element.

Internal set

In mathematical logic, in particular in model theory and non-standard analysis, an

internal set is a set that is a member of a model.

Internal set is the key tool in formulating the transfer principle, which concerns

the logical relation between the properties of the real numbers R, and the

properties of a larger field denoted *R called the hyperreals. The field *R

includes, in particular, infinitesimal ("infinitely small") numbers, providing a

rigorous mathematical realisation of a project initiated by Leibniz. Roughly

speaking, the idea is to express analysis over R in a suitable language of

mathematical logic, and then point out that this language applies equally well to

*R. This turns out to be possible because at the set-theoretic level, the

propositions in such a language are interpreted to apply only to internal sets

rather than to all sets (note that the term "language" is used in a loose sense in

the above).

Internal sets in the ultrapower construction

Relative to the ultrapowerconstruction of the hyperreal numbers as equivalence

classes of sequences , an internal subset [An] of *R is one defined by a

sequence of real sets , where a hyperreal [un] is said to belong to the set

if and only if the set of indices n such that , is a member of

the ultrafilter used in the construction of *R. This can be likened to the funfair

game "guess your weight," where someone guesses the contestant's weight, with
http://en.wikipedia.org/wiki/Convex_sethttp://en.wikipedia.org/wiki/Normalizing_constanthttp://en.wikipedia.org/wiki/Continuous_functionhttp://en.wikipedia.org/wiki/Mathematical_logichttp://en.wikipedia.org/wiki/Model_theoryhttp://en.wikipedia.org/wiki/Non-standard_analysishttp://en.wikipedia.org/wiki/Transfer_principlehttp://en.wikipedia.org/wiki/Ultrapowerhttp://en.wikipedia.org/wiki/Hyperreal_numberhttp://en.wikipedia.org/wiki/Ultrafilterhttp://en.wikipedia.org/wiki/Funfairhttp://en.wikipedia.org/wiki/Funfairhttp://en.wikipedia.org/wiki/Ultrafilterhttp://en.wikipedia.org/wiki/Hyperreal_numberhttp://en.wikipedia.org/wiki/Ultrapowerhttp://en.wikipedia.org/wiki/Transfer_principlehttp://en.wikipedia.org/wiki/Non-standard_analysishttp://en.wikipedia.org/wiki/Model_theoryhttp://en.wikipedia.org/wiki/Mathematical_logichttp://en.wikipedia.org/wiki/Continuous_functionhttp://en.wikipedia.org/wiki/Normalizing_constanthttp://en.wikipedia.org/wiki/Convex_set


34/54

closer guesses being more correct, and where the guesser "wins" if he or she

guesses near enough to the contestant's weight, with the actual weight being

completely correct (mapping to 1 by the membership function).

Fuzzy interval

A fuzzy interval is an uncertain set with a mean interval whose elements

possess the membership function value A(x) = 1. As in fuzzy numbers, the

membership function must be convex, normalized, at least segmentally

continuous.

Internal set theory

Internal set theory (IST) is a mathematical theory ofsets developed by Edward

Nelson that provides an axiomatic basis for a portion of the non-standard

analysis introduced by Abraham Robinson. Instead of adding new elements to

the real numbers, the axioms introduce a new term, "standard", which can be

used to make discriminations not possible under the conventional axioms for

sets. Thus, the starting point of IST is a modification of ZFC. In particular, non-

standard elements within the set of real numbers can be shown to have

properties that correspond to the properties of infinitesimal and unlimited

elements.

Nelson's formulation is made more accessible for the lay-mathematician by

leaving out many of the complexities of meta-mathematical logic that were

initially required to justify rigorously the consistency of infinitesimal elements.
http://en.wikipedia.org/wiki/Convex_sethttp://en.wikipedia.org/wiki/Normalizing_constanthttp://en.wikipedia.org/wiki/Continuous_functionhttp://en.wikipedia.org/wiki/Set_%28mathematics%29http://en.wikipedia.org/wiki/Edward_Nelsonhttp://en.wikipedia.org/wiki/Edward_Nelsonhttp://en.wikipedia.org/wiki/Non-standard_analysishttp://en.wikipedia.org/wiki/Non-standard_analysishttp://en.wikipedia.org/wiki/Abraham_Robinsonhttp://en.wikipedia.org/wiki/Real_numberhttp://en.wikipedia.org/wiki/Zermelo-Fraenkel_axiomshttp://en.wikipedia.org/wiki/Zermelo-Fraenkel_axiomshttp://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theoryhttp://en.wikipedia.org/wiki/Infinitesimalhttp://en.wikipedia.org/wiki/Logichttp://en.wikipedia.org/wiki/Logichttp://en.wikipedia.org/wiki/Infinitesimalhttp://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theoryhttp://en.wikipedia.org/wiki/Zermelo-Fraenkel_axiomshttp://en.wikipedia.org/wiki/Zermelo-Fraenkel_axiomshttp://en.wikipedia.org/wiki/Real_numberhttp://en.wikipedia.org/wiki/Abraham_Robinsonhttp://en.wikipedia.org/wiki/Non-standard_analysishttp://en.wikipedia.org/wiki/Non-standard_analysishttp://en.wikipedia.org/wiki/Edward_Nelsonhttp://en.wikipedia.org/wiki/Edward_Nelsonhttp://en.wikipedia.org/wiki/Set_%28mathematics%29http://en.wikipedia.org/wiki/Continuous_functionhttp://en.wikipedia.org/wiki/Normalizing_constanthttp://en.wikipedia.org/wiki/Convex_set


35/54

Intuitive justification

Whilst IST has a perfectly formal axiomatic scheme, described below, an intuitive

justification of the meaning of the term 'standard' is desirable. This is not part of

the official theory, but is a pedagogical device that might help the student engage

with the formalism. The essential distinction, similar to the concept of definable

numbers, contrasts the finiteness of the domain of concepts that we can specify

and discuss with the unbounded infinity of the set of numbers; compare finitism.

The number of symbols we write with is finite.

The number of mathematical symbols on any given page is finite.

The number of pages of mathematics a single mathematician can producein a lifetime is finite.

Any workable mathematical definition is necessarily finite.

There are only a finite number of distinct objects a mathematician can

define in a lifetime.

There will only be a finite number of mathematicians in the course of our

(presumably finite) civilisation.

Hence there is only a finite set of whole numbers our civilisation can

discuss in its allotted timespan.

What that limit actually is, is unknowable to us, being contingent on many

accidental cultural factors.

This limitation is not in itself susceptible to mathematical scrutiny, but the

fact that there is such a limit, whilst the set of whole numbers continues

forever without bound, is a mathematical truth.

The term standard is therefore intuitively taken to correspond to somenecessarily finite portion of "accessible" whole numbers. In fact the argument can

be applied to any infinite set of objects whatsoever - there are only so many

elements that we can specify in finite time using a finite set of symbols and there

are always those that lie beyond the limits of our patience and endurance, no
http://en.wikipedia.org/wiki/Definable_numberhttp://en.wikipedia.org/wiki/Definable_numberhttp://en.wikipedia.org/wiki/Finitismhttp://en.wikipedia.org/wiki/Finitismhttp://en.wikipedia.org/wiki/Definable_numberhttp://en.wikipedia.org/wiki/Definable_number


36/54

matter how we persevere. We must admit to a profusion of non-standard

elements too large or too anonymous to grasp within any infinite set.

Principles of the standardpredicate

The following principles follow from the above intuitive motivation and so should

be deducible from the formal axioms. For the moment we take the domain of

discussion as being the familiar set of whole numbers.

Any mathematical expression that does not use the new predicate

standardexplicitly or implicitly is a Classical Formula.

Any definition that does so is a Non-Classical Formula.

Any number uniquely specified by a classical formula is standard (by

definition).

Non-standard numbers are precisely those that cannot be uniquely

specified (due to limitations of time and space) by a classical formula.

Non-standard numbers are elusive: each one is too enormous to be

manageable in decimal notation or any other representation, explicit or

implicit, no matter how ingenious your notation. Whatever you succeed in

producing is by-definition merely another standard number. Nevertheless, there are (many) non-standard whole numbers in any

infinite subset ofN.

Non-standard numbers are completely ordinary numbers, having decimal

representations, prime factorisations, etc. Every classical theorem that

applies to the natural numbers applies to the non-standard natural

numbers. We have created, not new numbers, but a new method of

discriminating between existing numbers.

Moreover - any classical theorem that is true for all standard numbers is

necessarily true for all natural numbers. Otherwise the formulation "the

smallest number that fails to satisfy the theorem" would be a classical

formula that uniquely defined a non-standard number.


37/54

The predicate "non-standard" is a logically consistent method for

distinguishing large numbers - the usual term will be illimited. Reciprocals

of these illimited numbers will necessarily be extremely small real

numbers - infinitesimals. To avoid confusion with other interpretations of

these words, in newer articles on IST those words are replaced with the

constructs "i-large" and "i-small".

There are necessarily only finitely many standard numbers - but caution is

required: we cannot gather them together and hold that the result is a well-

defined mathematical set. This will not be supported by the formalism (the

intuitive justification being that the precise bounds of this set vary with time

and history). In particular we will not be able to talk about the largest

standard number, or the smallest non-standard number. It will be valid to

talk about some finite set that contains all sta

About Sets

Documents

Transcript of About Sets