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{{r|Set theory}}

A classic theorem of set theory asserting that sets can be ordered by size.

...h J. Devlin | authorlink=Keith Devlin | title=Fundamentals of Contemporary Set Theory | series=Universitext | publisher=[[Springer-Verlag]] | year=1979 | isbn=0- * {{cite book | author=Paul Halmos | authorlink=Paul Halmos | title=Naive set theory | series=The University Series in Undergraduate Mathematics | publisher=[[V

In [[set theory]], a '''subset''' of a [[set (mathematics)|set]] ''X'' is a set ''A'' whose

...veral possible formulations of [[Set_theory#Axiomatic_set_theory|axiomatic set theory]].

In [[set theory]], the '''characteristic function''' or '''indicator function''' of a [[sub

However, the term "aleph-0" is mainly used in the context of [[set theory]]; which finally turned out to be independent of the axioms of set theory:

...ages}}</noinclude>(1845-1918) Danish-German mathematician who introduced [[set theory]] and the concept of [[transcendental number]]s

...open set]]s are those which have [[countable set|countable]] [[complement (set theory)|complement]], together with the empty set. Equivalently, the [[closed set

...the [[open set]]s are those which have [[finite set|finite]] [[complement (set theory)|complement]], together with the empty set. Equivalently, the [[closed set

In [[set theory]], a '''pointed set''' is a [[set (mathematics)|set]] together with a disti

...ties that have the same structure as the [[Schröder-Bernstein theorem]] of set theory.

{{r|Set theory}} {{r|Complement (set theory)}}

{{r|descriptive set theory}}

* {{cite book | author=Paul Halmos | authorlink=Paul Halmos | title=Naive set theory | series=The University Series in Undergraduate Mathematics | publisher=[[V ...h J. Devlin | authorlink=Keith Devlin | title=Fundamentals of Contemporary Set Theory | series=Universitext | publisher=[[Springer-Verlag]] | year=1979 | isbn=0-

...h J. Devlin | authorlink=Keith Devlin | title=Fundamentals of Contemporary Set Theory | series=Universitext | publisher=[[Springer-Verlag]] | year=1979 | isbn=0- * {{cite book | author=Paul Halmos | authorlink=Paul Halmos | title=Naive set theory | series=The University Series in Undergraduate Mathematics | publisher=[[V

...h J. Devlin | authorlink=Keith Devlin | title=Fundamentals of Contemporary Set Theory | series=Universitext | publisher=[[Springer-Verlag]] | year=1979 | isbn=0- * {{cite book | author=Paul Halmos | authorlink=Paul Halmos | title=Naive set theory | series=The University Series in Undergraduate Mathematics | publisher=[[V

...h J. Devlin | authorlink=Keith Devlin | title=Fundamentals of Contemporary Set Theory | series=Universitext | publisher=[[Springer-Verlag]] | year=1979 | isbn=0- * {{cite book | author=Paul Halmos | authorlink=Paul Halmos | title=Naive set theory | series=The University Series in Undergraduate Mathematics | publisher=[[V

...h J. Devlin | authorlink=Keith Devlin | title=Fundamentals of Contemporary Set Theory | series=Universitext | publisher=[[Springer-Verlag]] | year=1979 | isbn=0- * {{cite book | author=Paul Halmos | authorlink=Paul Halmos | title=Naive set theory | series=The University Series in Undergraduate Mathematics | publisher=[[V

In [[set theory]], the '''intersection''' of two sets is the set of elements that they have * {{cite book | author=Paul Halmos | authorlink=Paul Halmos | title=Naive set theory | series=The University Series in Undergraduate Mathematics | publisher=[[V

In [[set theory]], a '''filter''' is a family of [[subset]]s of a given set which has prope ...bseteq X</math> either <math>A \in \mathcal{F}</math> or the [[complement (set theory)|complement]] <math>X \setminus A \in \mathcal{F}</math>.

requires advanced results from [[descriptive set theory]].

In [[set theory]], '''union''' (denoted as ∪) is a set operation between two sets that fo * {{cite book | author=Paul Halmos | authorlink=Paul Halmos | title=Naive set theory | series=The University Series in Undergraduate Mathematics | publisher=[[V

...h J. Devlin | authorlink=Keith Devlin | title=Fundamentals of Contemporary Set Theory | series=Universitext | publisher=[[Springer-Verlag]] | year=1979 | isbn=0- * {{cite book | author=Paul Halmos | authorlink=Paul Halmos | title=Naive set theory | series=The University Series in Undergraduate Mathematics | publisher=[[V

A characteristic property of finite sets (which, in [[set theory]] is used to ''define'' finite sets) is the following:

{{r|Inclusion (set theory)}}

{{r|Set theory}}

...s and [[Kurt Gödel]] showed that it was independent of the other axioms of set theory.

...e possible to take the concept of ordered pair as an elementary concept in set theory, but it is more usual to define them in terms of sets. Kuratowksi proposed ...h J. Devlin | authorlink=Keith Devlin | title=Fundamentals of Contemporary Set Theory | series=Universitext | publisher=[[Springer-Verlag]] | year=1979 | isbn=0-

...veral possible formulations of [[Set_theory#Axiomatic_set_theory|axiomatic set theory]]. {{cite book |title=Set theory |author=Thomas J Jech |url= http://books.google.com/books?id=pLxq0myANiEC&p

and initiated the study of infinite numbers, now a major branch of [[set theory]].

{{r|Set theory}}

| title = Classical descriptive set theory

{{r|Complement (set theory)}}

{{rpl|Fibre (set theory)}}

...rel; see [[Non-Borel_set/Advanced]] if you are acquainted with descriptive set theory. If you are not, you may find it instructive to try proving that ''A'' is B

...ins ''X'' itself and is closed under the operation of taking [[complement (set theory)|complement]]s, finite [[union]]s and finite [[intersection]]s in ''X''. Th

In [[set theory]], the '''kernel of a function''' is the [[equivalence relation]] on the do

{{r|Set theory}}

*In [[mathematics]], the term ''element'' is mainly used in [[set theory]], and in various combinations:

In [[set theory]], the '''symmetric difference''' of two sets is the set of elements that b

{{r|Set theory}}

...hen considered as a partially [[ordered set]] with respect to [[inclusion (set theory)|inclusion]].

{{r|Set theory}}

* {{cite book | author=Paul Halmos | authorlink=Paul Halmos | title=Naive set theory | series=The University Series in Undergraduate Mathematics | publisher=[[V ...h J. Devlin | authorlink=Keith Devlin | title=Fundamentals of Contemporary Set Theory | series=Universitext | publisher=[[Springer-Verlag]] | year=1979 | isbn=0-

{{r|Class (set theory)|In mathematics}}

...closed if <math>X-A=\{x \in X \mid x \notin A\}</math>, the [[complement (set theory)|complement]] of <math>A</math> in <math>X</math>, is an [[open set]]. The

In [[set theory]], a '''transitive relation''' on a [[set (mathematics)|set]] is a [[relati

* In [[set theory]], [[intersection]] and [[union]];

* {{cite book | author=Paul Halmos | authorlink=Paul Halmos | title=Naive set theory | series=The University Series in Undergraduate Mathematics | publisher=[[V

{{r|Set theory}}

{{r|Set theory}}

...ician '''Georg Ferdinand Ludwig Philipp Cantor''' (1845-1918) introduced [[set theory]] and the concept of [[transfinite number]]s.

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* In [[set theory]], ''standard model'' of the [[natural number]]s usually refers to the set

{{r|Set theory}}

{{r|Set theory}}

In [[set theory]], '''cardinality''' is a property of [[set]]s that generalises the notion ...the [[axiom of foundation]] holds, the set of all sets of minimal [[rank (set theory)|rank]] equinumerous with ''X'' can be used. If the [[axiom of choice]] is

The '''inclusion-exclusion principle''' is a [[theorem]] of [[set theory]] relating to the [[cardinality]] of a set defined as the [[union (mathemat

* {{cite book | author=Paul Halmos | authorlink=Paul Halmos | title=Naive set theory | series=The University Series in Undergraduate Mathematics | publisher=[[V

...hen considered as a partially [[ordered set]] with respect to [[inclusion (set theory)|inclusion]].

* {{cite book | author=Paul Halmos | authorlink=Paul Halmos | title=Naive set theory | series=The University Series in Undergraduate Mathematics | publisher=[[V

** In [[set theory]], [[intersection]] distributes over [[union]] and union distributes over i

* {{cite book | author=Paul Halmos | authorlink=Paul Halmos | title=Naive set theory | series=The University Series in Undergraduate Mathematics | publisher=[[V

In [[set theory]], '''composition''' is an operation on [[relation (mathematics)|relations]

* Zimmermann H., ''Fuzzy Set Theory and its Applications'' (2001), ISBN 978-0-7923-7435-0. * Klir G. , UTE H. St.Clair and Bo Yuan ''Fuzzy Set Theory Foundations and Applications'',1997.

...properties as ''"to be small"'', ''"to be close to 6"'' and so on. Now, in set theory given a set ''S'' and a "well defined" property ''P'', the ''axiom of comp ...erent empty subsets and therefore there is not a unique empty subset as in set theory.

{{r|Set theory}}

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...m of choice and of the generalized continuum-hypothesis with the axioms of set theory'') Paul J. Cohen, ''Set theory and the continuum hypothesis''. New York, Amsterdam. 1966.

...out to be a tricky matter. However, some unproblematic examples from naïve set theory will make the concept clearer. These examples will be used throughout this == History of set theory ==

...e notion of comparison between [[number]]s and magnitudes, or [[inclusion (set theory)|inclusion]] between sets or [[algebraic structure]]s. ...maximal within the family of chains ordered by set-theoretic [[inclusion (set theory)|inclusion]]).

...lved.) The very existence of various sets introduced below is addressed by set theory, for example by the [[Zermelo-Fraenkel axioms]].<ref name=Jech/> See {{cite book |title=Naive set theory |author=Paul Richard Halmos |chapter=Section 9: Families |url=http://books.

...mes '''Cantor-Schröder-Bernstein theorem''') is a fundamental theorem of [[set theory]]. * '''1895''' [[Georg Cantor]] states the theorem in his first paper on set theory and transfinite numbers (as an easy consequence of the linear order of card

...mes '''Cantor-Schröder-Bernstein theorem''') is a fundamental theorem of [[set theory]]. * '''1895''' [[Georg Cantor]] states the theorem in his first paper on set theory and transfinite numbers (as an easy consequence of the linear order of card

...out to be a tricky matter. However, some unproblematic examples from naïve set theory will make the concept clearer. These examples will be used throughout this == History of set theory ==

who showed that &ndash; in set theory including the [[axiom of choice]] &ndash; ...othesis is independent of the usual [[axiomatic set theory|(ZFC) axioms of set theory]].

...e subjects. For example, arithmetic has the product of a pair of numbers, set theory has the Cartesian product of a pair of sets and logic has the conjunction o

...b> is a subset of ''E''_''n''+1 for all ''n'', then the [[Union (set theory)|union]] of the sets ''E''_''n'' is measurable, and ...a subset of ''E''_''n'' for all ''n'', then the [[Intersection (set theory)|intersection]] of the sets ''E''_''n'' is measurable; furthermor

...s [[nitrogen]]", and unlike other explanations - realist or nominalist - [[set theory]] provides a mature understanding of classes including identity conditions.

...ann Benedict Listing]]. Modern topology depends strongly on the ideas of [[set theory]], developed by [[Georg Cantor]] in the later part of the 19th century. [[H

or, in set theory, as a specific set that serves as a concrete object (model) In modern mathematics, in particular because of set theory and

* Zimmermann H., ''Fuzzy Set Theory and its Applications'' (2001), ISBN 0-7923-7435-5.

In the set theory, infinity appears directly; for instance,

the sum being taken on ''E''₁ of the ''d'' points on the [[fibre (set theory)|fibre]] over ''Q''. This is indeed an isogeny, and the [[function composi

...alogy to the [[Schröder-Bernstein theorem|theorem]] of the same name (from set theory).

...alogy to the [[Schröder-Bernstein theorem|theorem]] of the same name (from set theory).

...defined object that underlies some [[Set_theory#The_paradoxes|paradoxes in set theory]]. The idea of a universe ''U'' need not be paradoxical, however, if one co

...Octonion]]s were discovered in 1843. [[Georg Cantor]], through its naive [[set theory]], formally defined the notion of [[infinity]] in 1895. [[Kurt Hensel]] fir

...>, we define the ''closed sets'' of <math>X</math> to be the [[complement (set theory)|complement]]s (in <math>X</math>) of the open sets; the closed sets of <ma