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  • Main Page
    * [[Set Theory]] * [[Measure Theory]]
    7 KB (999 words) - 16:51, 11 May 2020
  • Bijection
    ...(X)</math> where <math>\mathcal{P}(X)</math> denotes the [[Power set|power set]] of <math>X</math>) {{Definition|Set Theory|Elementary Set Theory}}
    732 B (124 words) - 11:49, 26 September 2016
  • Injection
    ...rjection/injection/[[bijection]] to be seen through the lens of [[Category Theory]]. [[User:Alec|Alec]] ([[User talk:Alec|talk]]) 21:50, 8 May 2018 (UTC) ...ijection where the cardinality is always 1 (and thus we take the singleton set <math>f^{-1}(y)=\{x\}</math> as the value it contains, writing {{M|1=f^{-1}
    3 KB (463 words) - 21:50, 8 May 2018
  • Surjection
    {{Definition|Set Theory}}
    2 KB (263 words) - 21:56, 8 May 2018
  • The relationship between logical implication and the subset relation
    ...n A\implies x\in B]</math><ref>Definition 3.10 (p10) - Introduction to Set Theory, Third Edition (Revised and Expanded) - Karel Hrbacek and Thomas Jech</ref> This is easily seen to be equivalent as if {{M|A}} is [[empty set|empty]] (so there is no {{M|x\in A}} to speak of) the implication is semant
    776 B (136 words) - 17:36, 16 January 2017
  • Basis for a topology
    ...but "let {{M|A\in\mathcal{P}(B)}}" instead. To emphasise that the [[power-set]] is possibly in play. ...se]], we usually deal with subsets of the ''space'' not subsets of the ''[[set system]]'' on that space.<br/>
    5 KB (802 words) - 18:35, 17 December 2016
  • The intersection of sets is a subset of each set
    {{Theorem Of|Set Theory}}
    820 B (159 words) - 21:35, 15 August 2015
  • Commutativity of intersection
    {{Theorem Of|Set Theory}}
    610 B (122 words) - 19:32, 28 October 2016
  • Measure space
    ...t, and {{M|\mathcal{A} }} is a [[Sigma-algebra|{{Sigma|algebra}}]] on that set (which together, as {{M|(X,\mathcal{A})}}, form a [[Measurable space|measur Given a set {{M|X}} and an [[Algebra of sets|algebra]], {{M|\mathcal{A} }} (NOT a {{sig
    1 KB (188 words) - 15:24, 21 July 2015
  • Function
    {{Definition|Set Theory}}
    4 KB (659 words) - 13:01, 19 February 2016
  • Set theory axioms
    ...sed to provide some discussion for the axioms (for example "there exists a set with no elements" doesn't really deserve its own page) ...Only "major" results are shown, they are covered in the motivation for set theory page, and "D" denotes "definition" - which is something that is natural to
    3 KB (619 words) - 10:25, 11 March 2015
  • Motivation for set theory axioms
    ...s. For example you are happy that {{M|\{1,2,3\}=\{2,1,3\} }} and that this set (I use singular because they are identical) contains the elements {{M|1}}, ...- for example we use a letter (<math>\emptyset</math>) to denote the empty set, what if there are 2 empty sets? We would like to justify this.
    3 KB (584 words) - 23:03, 28 February 2015
  • Relation
    ** For example {{M|<}} is a relation in the set of {{M|\mathbb{Z} }} (the integers) ! Set relation
    4 KB (762 words) - 20:07, 20 April 2016
  • Ordered pair
    The axioms may be found [[Set theory axioms|here]] ...know <math>\{\{a\}\}=\{\{a'\},\{a',b'\}\}</math> so again using the [[Set theory axioms]] (namely Extensionality) we see <math>a=a'=b'</math> so <math>a=a'<
    2 KB (327 words) - 07:22, 27 April 2015
  • The domain, range and field of a relation exist
    ...>\{x|\exists y:(x,y)\in R\}</math> exists, to do this we require the [[Set theory axioms|axioms]] of schema and union. {{Theorem Of|Set Theory}}
    339 B (63 words) - 07:22, 27 April 2015
  • Todo list
    | Proof that a function is invertible {{M|\iff}} it is one-to-one (Jech, Set Theory -p25) | Proof that <math>B^A</math> - the set of all functions from {{M|A}} into {{M|B}} exists
    464 B (76 words) - 16:54, 10 June 2015
  • Ordering
    {{Definition|Set Theory}}
    5 KB (1,006 words) - 13:21, 1 January 2016
  • Equivalence relation
    * An [[equivalence class]] is the name given to the set of all things which are equivalent under a given equivalence relation. **[[The equivalence classes of an equivalence relation partitions a set]].
    3 KB (522 words) - 15:18, 12 February 2019
  • Index of notation
    ! [[Index of set-like notations]] | set-like notations
    9 KB (1,490 words) - 06:13, 1 January 2017
  • Sequence
    ...>\{a_n\}_{n=1}^\infty</math> however I don't like this, as it looks like a set. I have seen the notation <math>(a_n)_{n=1}^\infty</math> and I must say I ...Maurin</ref>, <math>f:\mathbb{N}\rightarrow S</math> where {{M|S}} is some set. For a finite sequence it is simply <math>f:\{1,...,n\}\rightarrow S</math>
    2 KB (419 words) - 18:12, 13 March 2016
  • Formal logic language
    Formal logic plays an important role, especially so in set theory, but overall an important role. It is important to have a concrete understa
    2 KB (410 words) - 16:35, 9 March 2015
  • Group
    A group is a set {{M|G}} and an operation <math>*:G\times G\rightarrow G</math>, denoted <ma If the operation is obvious then "Let {{M|G}} be the set of (whatever) and let {{M|(G,+)}} be a group"
    7 KB (1,332 words) - 07:17, 16 October 2016
  • Cardinality
    Informally the cardinality of a set is the number of things in it. The cardinality of a set {{M|A}} is denoted <math>|A|</math>
    2 KB (327 words) - 10:25, 12 March 2015
  • Well-ordering
    An ordering <math><</math> of a set {{M|P}} which is both: ...{{M|A\ne\emptyset}} has a least element. (Then {{M|A}} is ''[[Well-ordered set|well-ordered]]''<ref name="Top">Topology - James R. Munkres - 2nd edition</
    488 B (76 words) - 17:34, 24 July 2015
  • Ring of sets
    Take a set {{M|X}}, the [[Power set|power set]] of {{M|X}}, {{M|\mathcal{P}(X)}} is a ring (further still, an [[Algebra o The empty set belongs to every ring
    2 KB (336 words) - 17:21, 18 August 2016
  • Algebra of sets
    * [[Types of set algebras]] {{Measure theory navbox|plain}}
    3 KB (507 words) - 18:43, 1 April 2016
  • Sigma-ring
    A non-empty class of sets {{M|S}} is a {{sigma|ring}} if<ref>Measure Theory, p24 - Halmos - Graduate Texts in Mathematics (18) - Springer</ref>: That is to say it is closed under [[Set subtraction|subtraction]] and [[Countable|countable]] [[Union|union]]
    728 B (125 words) - 15:34, 13 March 2015
  • Sigma-algebra
    # Show a {{sigma|algebra}} is closed under [[set-subtraction]], {{M|\forall A,B\in\mathcal{A}[A-B\in\mathcal{A}]}} * {{M|\mathcal{A} }} is closed under [[Set subtraction|set subtraction]]
    8 KB (1,306 words) - 01:49, 19 March 2016
  • Measure Theory
    ...generated by#Every set in R(A) can be finitely covered by sets in A|Every set in {{M|R(\mathcal{J}^n)}} can be finitely covered by sets in {{M|\mathcal{J But we do not know that every set in {{M|R(\mathcal{J}^n)}} can be finitely covered by DISJOINT sets in {{M|\
    4 KB (733 words) - 01:41, 28 March 2015
  • Index of terms
    * Measure Theory | Refers to the set <math>\mathbb{R}\cup\{+\infty,-\infty\}</math> where it is understood that
    409 B (62 words) - 15:49, 13 March 2015
  • Extended real value
    The set <math>\mathbb{R}\cup\{-\infty,+\infty\}</math> refers to "extended real val Halmos - Measure Theory - Page 1 - Spring - Graduate Texts in Mathematics (18)</ref>
    2 KB (396 words) - 16:07, 13 March 2015
  • Set function
    ...] with a domain of definition that is a class of sets<ref>Halmos - Measure Theory - p30 - Springer - Graduate Texts in Mathematics (18)</ref> {{Definition|Set Theory|Measure Theory}}
    355 B (54 words) - 16:10, 13 March 2015
  • Additive function
    {{Requires references|See Halmos' measure theory book too}} ...ve function (which way have meaning in say algebra), be sure to update the SET FUNCTION redirects that point into this page
    6 KB (971 words) - 18:16, 20 March 2016
  • Pre-measure
    *# {{MM|1=\mu_0(\emptyset)=0}} - the measure of the empty set is {{M|0}} * {{MM|1=\mu_0(\emptyset)=0}} - the measure of the empty set is {{M|0}} and
    5 KB (782 words) - 01:49, 26 July 2015
  • Measure
    A (positive) ''measure'', {{M|\mu}} is a [[set function]] from a [[sigma-ring|{{sigma|ring}}]], {{M|\mathcal{R} }}, to the ...n\right)=\sum_{n=1}^\infty\mu(A_n)]}} ({{M|\mu}} is a [[countably additive set function]])
    6 KB (941 words) - 14:39, 16 August 2016
  • Measurable space
    {{Refactor notice|grade=A*|msg=Lets get this measure theory stuff sorted. At least the skeleton Given a [[set]], {{M|X}}, and a [[sigma-algebra|{{sigma|algebra}}]], {{M|\mathcal{A}\in\m
    2 KB (248 words) - 13:05, 2 February 2017
  • Lebesgue measure
    The set function <math>\lambda^n:(\mathbb{R}^n,\mathcal{B}(\mathbb{R}^n))\rightarro Where <math>\mathcal{J}=</math> the set of all half-open-half-closed 'rectangles' in <math>\mathbb{R}^n</math>
    804 B (129 words) - 00:28, 20 December 2016
  • Semi-ring of sets/Definition
    A collection of {{plural|set|s}}, {{M|\mathcal{F} }}<ref group="Note">An F is a bit like an R with an un ...mS_i}}. We require that they be pairwise disjoint AND their union be the [[set difference]] of {{M|S}} and {{M|T}}.</ref> - this doesn't require {{M|S-T\i
    2 KB (337 words) - 17:25, 18 August 2016
  • Ring generated by
    ===Every set in R(A) can be finitely covered by sets in A=== If {{M|A}} is any class of sets, then every set in {{M|R(A)}} can be covered by a finite union of sets in {{M|A}}
    2 KB (307 words) - 07:24, 27 April 2015
  • Complement
    The complement of a set is everything not in it. For example given a set {{M|A}} in a space {{M|X}} the complement of {{M|A}} (often denoted {{M|A^c It may also be written using [[Set subtraction|set subtraction]]
    726 B (145 words) - 13:28, 18 March 2015
  • Ring
    A set {{M|R}} and two [[Binary operation|binary operations]] {{M|+}} and {{M|\tim {{Definition|Abstract Algebra|Ring Theory}}
    7 KB (1,248 words) - 05:02, 16 October 2016
  • Monotonic set function
    Sometimes called '''Monotone set function'''. A set function {{M|f:E\rightarrow [0,\infty]\subset\mathbb{R} }} is '''monotonic'
    548 B (87 words) - 14:18, 18 March 2015
  • Subtractive set function
    ...{R}\cup\{-\infty,\infty\}</math> is subtractive<ref>p37 - Halmos - Measure Theory - Graduate Texts in Mathematics - 18</ref> if whenever {{M|A,B\in E}} we ha * [[Monotonic set function]]
    409 B (71 words) - 14:24, 18 March 2015
  • Measures are monotonic and subtractive
    ...]] and [[Subtractive set function|subtractive]]<ref>p37 - Halmos - Measure Theory - Graduate Texts in Mathematics - book 18</ref> {{Theorem Of|Measure Theory}}
    318 B (41 words) - 07:26, 27 April 2015
  • Probability space
    | Set ...s our ''state'' or ''sample'' space - I prefer ''state'' because it is the set of states the experiment may take.
    2 KB (338 words) - 22:55, 2 May 2015
  • Equivalence class
    {{Definition|Set Theory|Abstract Algebra}}
    1 KB (160 words) - 20:00, 14 November 2015
  • The fundamental group
    {{Definition|Topology|Homotopy Theory}} * <math>\pi_1(X,x_0)</math> denotes the set of [[Homotopy class|homotopy classes]] of [[Paths and loops in a topologica
    3 KB (393 words) - 16:10, 4 November 2016
  • Almost always
    {{Todo|But in measure theory and probability it means all but a set of measure zero}} *: The set {{M|\{x\vert f(x)\ge 10\} }} is finite (assuming that ''f'' runs over natur
    694 B (115 words) - 21:44, 19 March 2016
  • Greatest common divisor
    ...of {{M|a}} and {{M|b}}<ref name="Crypt">The mathematics of ciphers, Number theory and RSA cryptography - S. C. Coutinho</ref> is the greatest positive intege ...ximum element, and the maximum is the same as the {{M|\text{Sup} }} of the set) we can state the {{M|\text{gcd} }} as follows:
    1 KB (252 words) - 08:33, 21 May 2015
  • Passing to the quotient (function)
    Suppose that {{M|1=w^{-1}(v)}} is [[empty-set|empty]] or contains 2 (or more!) elements, then what do we define {{M|\tild {{Theorem Of|Elementary Set Theory|Set Theory}}
    8 KB (1,644 words) - 20:49, 11 October 2016
  • Indicator function
    {{Definition|Set Theory}}
    600 B (91 words) - 02:26, 7 June 2015
  • Types of set algebras
    ==Measure theory perspective== ...e class of sets {{M|\mathcal{A} }} is a collection of subsets from another set {{M|\Omega}}
    3 KB (449 words) - 20:06, 19 August 2016
  • Set
    * See [[Notes:Set]] {{Definition|Set Theory}}
    60 B (9 words) - 22:28, 19 August 2016
  • Semi-ring of half-closed-half-open intervals
    ...]] of [[set|sets]] where one or more of the {{M|X_\alpha}} are the [[empty set]], {{M|\emptyset}}, then: {{Measure theory navbox|plain}}
    4 KB (680 words) - 00:23, 20 August 2016
  • Index of properties
    | {{M|\backslash}}-closed<ref name="PTACC">Probability Theory - A comprehensive course - Second Edition - Achim Klenke</ref> ...{{M|\backslash}}-closed uses {{M|\backslash}} to denote [[Set subtraction|set subtraction]]<ref group="Note">This is because {{M|-}}-closed is not a good
    2 KB (360 words) - 20:43, 15 June 2015
  • Class of sets closed under complements properties
    ...a system of subsets of {{M|\Omega}} such that<ref name="PTACC">Probability Theory - A comprehensive course - Achim Klenke</ref>: * [[Types of set algebras]]
    1 KB (165 words) - 20:50, 15 June 2015
  • Properties of classes of sets closed under set-subtraction
    Suppose {{M|\mathcal{A} }} is an arbitrary class of [[set|sets]] with the property that: ...=\forall A,B\in\mathcal{A}[A-B\in\mathcal{A}]}} where {{M|A-B}} denotes "[[set subtraction]]" ({{AKA}}: [[relative complement]])
    3 KB (490 words) - 11:38, 21 August 2016
  • De Morgan's laws
    {{Theorem Of|Set Theory}}
    364 B (69 words) - 13:14, 16 June 2015
  • Sigma-algebra generated by
    ...set|power set]] of {{M|\Omega}}) there exists<ref name="PTACC">Probability Theory - A Comprehensive Course - Second Edition - Achim Klenke</ref> a smallest [ #* By [[The intersection of sets is a subset of each set]] we now see that {{M|\sigma(S)\subseteq\mathcal{P}(\Omega)}}
    2 KB (286 words) - 22:02, 17 March 2016
  • Partition (abstract algebra)
    ...ry [[Indexing set|indexing set]], to each element {{M|i\in I}} we assign a set {{M|A_i}} which is non-empty. If: ...Analysis - Part 1: Elements - Krzystof Maurin</ref> ''the partition of the set {{M|B}} into classes {{M|A_i}} for {{M|i\in I}}''
    2 KB (316 words) - 16:03, 18 June 2015
  • Site projects:Patrolling measure theory/Task list
    * [[Class of sets closed under set-subtraction properties]] - '''DONE''' [[User:Alec|Alec]] ([[User talk:Alec| * [[Integral (measure theory)]] '''DONE''' [[User:Alec|Alec]] ([[User talk:Alec|talk]]) 02:01, 19 March
    5 KB (645 words) - 11:40, 21 August 2016
  • Union of subsets is a subset of the union
    * [[The intersection of sets is a subset of each set]] {{Theorem Of|Set Theory}}
    2 KB (290 words) - 21:33, 15 August 2015
  • Implies-subset relation
    {{Definition|Set Theory}}
    165 B (23 words) - 21:00, 12 May 2016
  • Group action
    A (left) ''group action'' of a [[group]] {{M|(G,*)}} on a [[set]] {{M|X}} is a [[mapping]]{{rAAPAG}}: ** [[The symmetric group on a set acts on the set by evaluation]]
    2 KB (320 words) - 23:28, 21 July 2016
  • Dynkin system generated by
    Given a set {{M|X}} and another set {{M|\mathcal{G}\subseteq\mathcal{P}(X)}} which we shall call the ''generato * [[Types of set algebras]]
    2 KB (245 words) - 15:16, 16 December 2016
  • Well-ordered set
    A set {{M|A}} with an [[Linear ordering|linear ordering]] {{M|<\subseteq A\times {{Definition|Set Theory}}
    916 B (164 words) - 17:49, 24 July 2015
  • Dynkin system
    * [[Types of set algebras]] {{Definition|Measure Theory}}
    1 KB (184 words) - 01:54, 19 March 2016
  • A collection of subsets is a sigma-algebra iff it is a Dynkin system and closed under finite intersections
    * A collection of subsets of a [[set]] {{M|X}}, say {{M|\mathcal{A} }}, is a [[Sigma-algebra|{{sigma|algebra}}]] {{Theorem Of|Measure Theory}}
    1 KB (239 words) - 13:21, 17 December 2016
  • Dynkin system/Definition 1
    <noinclude>{{Extra Maths}}</noinclude>Given a set {{M|X}} and a family of subsets of {{M|X}}, which we shall denote {{M|\math {{Definition|Measure Theory}}
    556 B (92 words) - 01:52, 19 March 2016
  • Dynkin system/Definition 2
    <noinclude>{{Extra Maths}}</noinclude>Given a set {{M|X}} and a family of subsets of {{M|X}} we denote {{M|\mathcal{D}\subset {{Definition|Measure Theory}}
    719 B (120 words) - 23:22, 2 August 2015
  • Function terminology
    ...ion {{M|f:A\rightarrow B}} where {{M|A}} and {{M|B}} are assumed to be any set unless stated otherwise we say the following about {{M|f}}: {{Definition|Abstract Algebra|Set Theory|Measure Theory}}
    1 KB (165 words) - 02:07, 3 August 2015
  • Min/max
    {{Definition|Set Theory|Real Analysis|Measure Theory}}
    884 B (144 words) - 02:51, 3 August 2015
  • Positive and negative parts of a function
    {{Definition|Real Analysis|Measure Theory|Set Theory}}
    518 B (84 words) - 01:21, 5 August 2015
  • Borel sigma-algebra generated by
    ...used, however in measure theory this notation is often used to denote the set of half-open-half-closed rectangles in {{M|\mathbb{R}^n}} - a totally separ {{Definition|Measure Theory}}
    2 KB (244 words) - 08:30, 6 August 2015
  • Borel sigma-algebra
    ...illing</ref>. It is [[Sigma-algebra generated by|generated by]] the [[Open set|open sets]] of the [[Metric space|metric space]] {{M|(\mathbb{R},\vert\cdot ...">Conventionally, {{M|\mathcal{J} }} denotes the open sets, but in measure theory this seems to denote the sets of half-open-half-closed rectangles, and it i
    5 KB (854 words) - 09:25, 6 August 2015
  • Map
    {{Definition|Set Theory}}
    48 B (5 words) - 15:42, 17 August 2015
  • Mapping
    {{Definition|Set Theory}}
    48 B (5 words) - 15:42, 17 August 2015
  • Correspondence
    {{Definition|Set Theory}}
    48 B (5 words) - 16:30, 23 August 2015
  • Outer-measure/Definition
    </noinclude>An ''outer-measure'', {{M|\mu^*}} is a [[set function]] from a [[hereditary sigma-ring|hereditary {{sigma|ring}}]], {{M| * an ''[[extended real valued]]'' [[countably subadditive set function]] that is [[monotonic]] and non-negative with the property: {{M|1=
    937 B (136 words) - 21:46, 8 April 2016
  • The (pre-)measure of a set is no more than the sum of the (pre-)measures of the elements of a covering for that set
    ...than the sum of the (pre-)measures of the elements of a covering for that set/Statement|Statement]]== ...than the sum of the (pre-)measures of the elements of a covering for that set/Statement}}
    4 KB (688 words) - 21:03, 31 July 2016
  • Limit of increasing sequence of sets
    {{Definition|Set Theory|Measure Theory|Real Analysis}}
    2 KB (334 words) - 17:52, 21 August 2015
  • Set subtraction
    Given two sets, {{M|A}} and {{M|B}} we define ''set subtraction'' ({{AKA}}: ''relative complement''{{rMTH}}) as follows: ==Trivial expressions for set subtraction==
    1 KB (237 words) - 00:48, 21 March 2016
  • Category
    # For every ordered pair, {{M|(X,Y)}} of ''objects'' a set {{M|\hom(X,Y)}} of ''morphisms'' {{M|f}} {{Definition|Category Theory}}
    2 KB (347 words) - 00:36, 27 September 2015
  • Classes of continuously differentiable functions
    Given {{M|U\subseteq\mathbb{R}^n}} (where {{M|U}} is [[Open set|open]]) and some {{M|k\ge 0}}, a function of the form: ...''"...functions of class {{M|C^k}} on {{M|U}} by..."'' when describing the set {{M|C^k(U)}}</ref> we require<ref name="ITSM"/>:
    3 KB (632 words) - 20:32, 16 October 2015
  • Real-valued function
    ...Lee - Springer GTM</ref>. That is to say any function ( {{M|f}} ) and any set ( {{M|U}} ) such that: {{Definition|Measure Theory|Manifolds|Differential Geometry|Functional Analysis}}
    807 B (117 words) - 23:15, 21 October 2015
  • Class of smooth real-valued functions on R-n
    denotes the set of all functions, {{M|:U\rightarrow\mathbb{R} }} that are [[smooth]] on {{M {{Definition|Manifolds|Functional Analysis|Real Analysis|Measure Theory}}
    2 KB (259 words) - 23:41, 21 October 2015
  • Pre-measure/New page
    ...ive) pre-measure'' is an ''[[extended real valued]]'' [[countably additive set function]], {{M|\bar{\mu}:\mathcal{R}\rightarrow\overline{\mathbb{R}_{\ge 0 * [[Types of set algebras]]
    3 KB (422 words) - 21:25, 17 August 2016
  • Subsequence
    {{Definition|Real Analysis|Set Theory|Functional Analysis}}
    2 KB (356 words) - 23:27, 16 November 2016
  • Cover
    ...of sets, {{M|1=\{S_\alpha\}_{\alpha\in I} }} is said to ''cover'' another set, {{M|X}} if{{rITTGG}}: {{Definition|Set Theory|Topology|Metric Space|Abstract Algebra}}
    415 B (63 words) - 11:02, 1 December 2015
  • Sub-sequence
    {{Definition|Set Theory|Real Analysis|Functional Analysis}}
    104 B (11 words) - 15:14, 1 December 2015
  • Subsequence/Definition
    {{Definition|Set Theory|Real Analysis|Functional Analysis}}
    3 KB (484 words) - 21:54, 16 November 2016
  • Cartesian product
    Given two sets, {{M|X}} and {{M|Y}} their ''Cartesian product'' is the set: ===Set construction===
    2 KB (318 words) - 14:25, 19 February 2016
  • Union
    {{Definition|Set Theory}}
    368 B (56 words) - 12:15, 16 March 2016
  • Partial ordering
    ...for anything other than denoting [[subset|subsets]], the relation and the set it relates on will go together, so you'll already be using {{M|\subseteq}} A tuple consisting of a set {{M|X}} and a partial order {{M|\sqsubseteq}} in {{M|X}} is called a [[pose
    4 KB (740 words) - 10:11, 20 February 2016
  • Strict partial ordering
    ...for anything other than denoting [[subset|subsets]], the relation and the set it relates on will go together, so you'll already be using {{M|\subseteq}} {{Order theory navbox|plain}}
    3 KB (436 words) - 10:15, 20 February 2016
  • Quotient (function)
    {{Definition|Set Theory|Abstract Algebra}}
    91 B (11 words) - 18:31, 8 October 2016
  • Overview of partial orders
    A ''partial order'' is a [[relation]] on a set {{M|X}}, which we shall call {{M|\mathcal{R}\subseteq X\times X}} that is{{ {{Order theory navbox|plain}}
    3 KB (454 words) - 07:40, 11 April 2016
  • Rational numbers
    {{Definition|Set Theory}}
    49 B (5 words) - 13:27, 8 January 2016
  • Index of notation/P
    * {{M|\mathbb{P} }} to denote [[set of predicates]] in [[order theory]] ...have seen no other meaning for {{M|\mathcal{P}(X)}} (where {{M|X}} is a [[set]]) however I have seen the notation:
    1 KB (184 words) - 23:58, 11 January 2016
  • SET (category)
    ...ts]] and every [[function]] (in the conventional sense, as mappings from 1 set to another) between those sets as the [[arrows of a category|arrows of the * '''Note: ''' sometimes the {{M|\mathrm{SET} }} category is {{AKA}} {{M|\mathrm{SETS} }} (and the page <code>[[SETS (ca
    1 KB (168 words) - 10:05, 19 February 2016
  • Category Theory (subject)
    '''Category theory is the study of objects linked by arrows, where arrows compose'''. In fact ...most familiar category to the reader will be [[SET (category)|{{M|\mathrm{SET} }}]]
    2 KB (311 words) - 11:46, 19 February 2016
  • SETS (category)
    #REDIRECT [[SET (category)]] {{Definition|Category Theory}}
    124 B (13 words) - 10:01, 19 February 2016
  • Site projects:Refactoring subjects
    ...coupled with some functions that have a property for all elements on that set, really falls under abstract algebra.
    3 KB (469 words) - 11:31, 19 February 2016
  • Site projects:Current projects
    * [[Site projects:Split set theory into elementary set theory]] ====Category Theory====
    868 B (93 words) - 12:15, 9 April 2016
  • Site:Mathematical subject index/Index
    ** [[Group Theory (subject)|Group Theory]] (see Group Theory for branches) ** [[Order Theory (subject)|Order Theory]] (see Order Theory for branches)
    2 KB (265 words) - 14:53, 26 February 2016
  • Poset
    A ''poset'' or ''partially ordered set'' is a [[tuple]] consisting of a [[set]], {{M|X}} and a [[partial ordering]], {{M|\preceq}}, on {{M|X}}, then: {{Requires references|I know Introduction to Category Theory has the definition, Analysis (Maurin) might too!}}
    421 B (60 words) - 21:24, 19 April 2016
  • Partial order
    {{Definition|Set Theory|Order Theory|Abstract Algebra}}
    87 B (10 words) - 09:31, 20 February 2016
  • Strict partial order
    {{Definition|Set Theory|Abstract Algebra|Order Theory}}
    94 B (11 words) - 09:40, 20 February 2016
  • Preorder
    A ''preorder'', {{M|\preceq}}, on a set {{M|X}} is a [[relation]] in {{M|X}}, so {{M|\preceq\subseteq X\times X}}, A tuple, consisting of a set {{M|X}}, equipped with a preorder {{M|\preceq}} is called a ''[[preset]]''<
    2 KB (355 words) - 10:13, 20 February 2016
  • Preset
    A ''preset'' is a [[tuple]] consisting of a [[set]] {{M|X}} and a [[preorder]] on {{M|X}}, {{M|\preceq}}{{rAITCTHS2010}}, the * [[Poset]] - the term for a set equipped with a [[partial ordering]] on itself.
    436 B (66 words) - 16:54, 1 March 2016
  • Preordering
    {{Definition|Order Theory|Set Theory|Abstract Algebra}}
    79 B (9 words) - 10:14, 20 February 2016
  • Order Theory (subject)
    ...from various kinds of orderings, called [[Lattice Theory (subject)|lattice theory]]. Some order theory is desired for parts of [[Analysis (subject)|analysis]], for this I recomme
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  • Exists functor
    ...] taking [[SET (category)|{{M|\mathrm{SET} }}]] {{M|\leadsto}} {{M|\mathrm{SET} }} defined as follows{{rAITCTHS2010}}: ...A\mapsto\mathcal{P}(A)]}}, recall {{M|\mathcal{P}(X)}} denotes the [[power set]] of {{M|X}}
    2 KB (317 words) - 17:51, 13 March 2016
  • Upper section
    ...r I am dealing with [[preset|presets]] not [[poset|posets]] here, so upper set might only be for posets, and upper section for presets, or both. Not sure * [[Lower section]] - the [[dual (order theory)|dual]] concept to this
    1 KB (171 words) - 16:35, 20 February 2016
  • Index of operators on ordered sets
    If you are given a set, say {{M|X}} and any of a: on that set, then this page indexes various operators that might take such a structured
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  • TOP (category)
    ...ological space|topological spaces]], the objects are [[tuple|tuples]] of a set {{M|X}} and a topology {{M|\mathcal{J}_X}} on {{M|X}} and the arrows, or mo {{Todo|Discuss as a subcategory of {{M|\mathrm{SET} }}, remember it must first go under the [[forgetful functor]] to discard t
    971 B (139 words) - 20:10, 20 February 2016
  • Measure Theory (subject)
    ...engths, so given a set where you can do these things (subtract and add - [[set subtraction]] and [[union]] respectively) you expect to be able to define a The measure theory project contains:
    832 B (121 words) - 15:24, 26 February 2016
  • Binary relation
    {{Definition|Set Theory}}
    48 B (5 words) - 18:42, 18 March 2016
  • Predicate
    ...dicate'', {{M|P}}, is a [[n-place relation|{{M|1}}-place relation]] on a [[set]] {{M|X}}<ref group="Note">{{M|P\subseteq X}} in this case. In contrast to ...comprehension]] - This states that given a set {{M|A}} we can construct a set {{M|B}} such that {{M|1=B=\{x\in A\ \vert P(x)\} }} for some ''predicate''
    916 B (160 words) - 18:44, 18 March 2016
  • Sigma-algebra/Definition
    </noinclude>Given a [[set]] {{M|X}} a ''{{sigma|algebra}} on {{M|X}}'' is a family of subsets of {{M| {{Definition|Measure Theory}}
    779 B (122 words) - 01:25, 19 March 2016
  • Notes:Measures
    ...thors to see what is what. Bogachev for example (author of [[Books:Measure Theory - Volume 1 - V. I. Bogachev]]) doesn't require that a measure even be posit # [[Books:Measure Theory - Volume 1 - V. I. Bogachev]]
    5 KB (771 words) - 03:06, 21 March 2016
  • Limsup and liminf
    {{Definition|Set Theory}}
    187 B (28 words) - 22:17, 19 April 2016
  • Limsup and liminf (sequence of sets)
    Given a [[set]] {{M|X}} and a [[sequence]] {{M|1=(A_n)^\infty_{n=1} }} of [[subset|subset ** In words: The superior limit of {{M|(A_n)}} is the set that contains {{M|x\in X}} given that {{M|x}} is in (countably) infinitely
    2 KB (386 words) - 22:17, 19 April 2016
  • Limsup (sequence of sets)
    {{Definition|Set Theory|Measure Theory}}
    99 B (14 words) - 00:36, 21 March 2016
  • Liminf (sequence of sets)
    {{Definition|Set Theory|Measure Theory}}
    99 B (14 words) - 00:35, 21 March 2016
  • Set complement
    {{Definition|Set Theory}} [[Category:Set operations]]
    79 B (8 words) - 00:43, 21 March 2016
  • Symmetric difference
    Let {{M|A,B\in\mathcal{P}(X)}} be two [[subset|subsets]] of a [[set]] {{M|X}}. We define the ''symmetric difference'' of {{M|A}} and {{M|B}} as ...A\triangle B:=(A-B)\cup(B-A)}}<ref group="Note">Here {{M|A-B}} denotes ''[[set subtraction]]''.</ref>
    830 B (139 words) - 00:59, 21 March 2016
  • Complementation
    {{Definition|Set Theory|Measure Theory}} [[Category:Set operations]]
    94 B (10 words) - 01:53, 21 March 2016
  • Notes:Halmos measure theory skeleton
    The "core" of Halmos' measure theory book is the following: # [[Measure]], {{M|\mu}}, countably additive extended real valued set function on a {{sigma|ring}}, {{M|\mathcal{R} }}, {{M|\mu:\mathcal{R}\right
    4 KB (581 words) - 20:31, 3 April 2016
  • Notes:Measure theory plan
    This document is the ''plan'' for the measure theory notation and development on this site. ...mu} }}) - Introduce a (positive) extended real valued [[countably additive set function]], {{M|\bar{\mu} }} on that ring. This will be a pre-measure and t
    4 KB (619 words) - 19:28, 24 May 2016
  • Site projects:Measure theory plan
    Inline with the [[Notes:Measure theory plan]] page we will use the terms and symbols which may be found on the bot * A page (perhaps [[measure theory and sigma algebras notice]] will need to (have part of it) be transcluded o
    2 KB (309 words) - 15:31, 27 March 2016
  • Notes:Quotient topology plan
    ...e varied. It's also not as if the concepts are even ''distinct'', I have a theory that given one form you actually induce the others. Thus there's really onl Thus the set one is certainly true.
    760 B (125 words) - 00:32, 22 April 2016
  • Measure theory terminology doctrine
    ...alued]]'' [[countably additive set function]] defined on a [[ring (measure theory)|ring]]. ...alued]]'' [[countably subadditive set function|countably '''sub'''additive set function]] defined on a [[hereditary sigma ring|hereditary {{sigma|ring}}]]
    2 KB (232 words) - 03:57, 30 March 2016
  • Notes:Halmos measure theory skeleton/Old page
    ==Halmos' measure theory: skeleton== * [[additive set function]]
    4 KB (674 words) - 19:46, 3 April 2016
  • Hereditary
    ...l{P}(A)[B\in\mathcal{H}])}} where {{M|\mathcal{P}(S)}} denotes the [[power set]] of {{M|S}}. * [[Hereditary set]] - a set that only contains other sets
    376 B (62 words) - 16:37, 22 August 2016
  • Hereditary system of sets
    {{Stub page|Needs linking to where it is used, notes on a sort of "power-set" like construct.|grade=B}} ...in [[Measure Theory (subject)|Measure Theory]] and ''[[Hereditary (measure theory)]]'' redirects here
    793 B (125 words) - 21:25, 19 April 2016
  • Hereditary sigma-ring
    ...rties. Additionally the "use" section requires expansion. Comment on power-set and sigma-algebra special case. Find out about related term, {{sigma|ideal} ...{{M|\mathcal{H} }}, is a system of sets that is both [[hereditary (measure theory)|hereditary]] and a [[sigma-ring|{{sigma|ring}}]]{{rMTH}}. This means {{M|\
    1 KB (220 words) - 21:25, 19 April 2016
  • Monotone
    {{Definition|Measure Theory|Order Theory|Set Theory}}
    77 B (9 words) - 19:35, 8 April 2016
  • Monotonic
    ...ires references|Find an order theory book, also I think that huge category theory PDF (Harold Simmons) has it}} # Unite with [[monotonic set function]]
    1 KB (190 words) - 04:50, 9 April 2016
  • Extension
    {{Definition|Set Theory}}[[Category:Function Terminology]]
    575 B (89 words) - 20:02, 8 April 2016
  • Restriction
    ...l{P}(X)}}<ref group="Note">Recall {{M|\mathcal{P}(X)}} denotes the [[power-set]] of {{M|X}}</ref> (so {{M|A\subseteq X}} - and is any subset) we define a {{Definition|Set Theory}}[[Category:Function Terminology]]
    652 B (107 words) - 20:01, 8 April 2016
  • Power-set
    #REDIRECT [[Power set]] {{Definition|Set Theory}}
    49 B (6 words) - 20:01, 8 April 2016
  • Extending pre-measures to outer-measures
    ...infty A_n\right\} }} - here {{M|\text{inf} }} denotes the [[infimum]] of a set. ...than the sum of the (pre-)measures of the elements of a covering for that set]], which states, symbolically:
    11 KB (1,921 words) - 16:59, 17 August 2016
  • Site projects:Split set theory into elementary set theory
    ...this way pages tagged as elementary set theory have no business in the set theory category. * Create an "elementary set theory" subject in the [[Site:Mathematical subject index]]
    739 B (114 words) - 12:21, 9 April 2016
  • Greater than or equal to
    {{Order theory navbox|plain}} {{Definition|Real Analysis|Elementary Set Theory|Order Theory}}
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  • Greater than or equal to/Epsilon form
    {{Theorem Of|Order Theory|Elementary Set Theory|Real Analysis}}
    1 KB (249 words) - 15:07, 9 April 2016
  • Infimum
    ...athcal{P}(X)}} where {{M|\mathcal{P}(S)}} denotes the [[power set]] of a [[set]] {{M|S}}</ref>. The ''infimum'' ({{AKA}}: ''greatest lower bound'', ''g.l. ...ce{\left\{x\in X\ \vert\ (\forall a\in A[x\preceq a])\right\} }_{\text{the set of all lower bounds of }A }\Big[b\preceq\text{Inf}(A)\Big]}} - which states
    5 KB (851 words) - 08:55, 29 July 2016
  • Reflexive relation
    {{Definition|Set Theory}}
    94 B (11 words) - 20:05, 20 April 2016
  • Transitive relation
    {{Definition|Set Theory}}
    95 B (11 words) - 20:06, 20 April 2016
  • Symmetric relation
    {{Definition|Set Theory}}
    94 B (11 words) - 20:07, 20 April 2016
  • Surjective
    {{Definition|Set Theory}}
    50 B (5 words) - 15:20, 21 April 2016
  • Passing to the quotient
    {{Definition|Topology|Set Theory}}
    136 B (15 words) - 12:49, 27 April 2016
  • Bijective
    {{Definition|Set Theory}}
    49 B (5 words) - 20:55, 2 May 2016
  • Subset
    A ''subset'', {{M|A}}, of a [[set]] {{M|B}} is a set that is entirely contained in {{M|B}}. That means everything contained in { {{Definition|Set Theory}}
    537 B (84 words) - 20:58, 12 May 2016
  • Upper bound
    * [[Supremum]] - the ''lowest'' upper bound of a set. * [[Lower bound]] - the [[dual (order theory)|dual]] concept.
    813 B (140 words) - 07:23, 20 May 2016
  • Lower bound
    * [[Infimum]] - the ''greatest'' lower bound of a set. * [[Upper bound]] - the [[dual (order theory)|dual]] concept.
    816 B (140 words) - 07:23, 20 May 2016
  • Notes:First order language
    ## Equality symbol - the set containing {{M|\doteq}}<ref name="WL"/><ref group="Note">This is done so th # Non-logical symbols<ref name="Both"/> (which vary from theory to theory<ref name="Both"/>), there are 3 kinds<ref name="WL"/>
    6 KB (1,088 words) - 09:22, 28 August 2016
  • Notes:Outer-measures to measures
    * {{M|\bar{S} }} - the set of all outer-measurable sets, WRT {{M|\mu^*}} (for the definition of outer- ...f he defines {{M|E_n}} as being a term "{{M|E_n}}" that happens to be in a set.... it's iffy at best (then {{M|E_j}} wouldn't make sense)
    6 KB (1,067 words) - 22:19, 23 May 2016
  • Hereditary sigma-ring generated by
    ...|grade=A*|msg=Needed for progress, I started the page to get some notation set in stone.}} {{Measure theory navbox|plain}}
    877 B (138 words) - 19:24, 24 May 2016
  • Mu*-measurable set
    #REDIRECT [[Outer splicing set]] {{Definition|Measure Theory}}
    62 B (7 words) - 22:05, 20 August 2016
  • The set of all mu*-measurable sets is a ring
    {{DISPLAYTITLE:The set of all {{M|\mu^*}}-measurable sets is a ring}}{{Stub page|grade=A*}} {{M|\mathcal{S} }}, [[The set of all mu*-measurable sets|the set of all {{M|\mu^*}} measurable sets]], is a [[ring of sets]]{{rMTH}}.
    8 KB (1,271 words) - 08:36, 29 May 2016
  • The set of all mu*-measurable sets is a sigma-ring
    {{Stub page|grade=A*|msg=Currently in the notes stage, see [[Notes:The set of all mu*-measurable sets is a ring]]}} ...0} }} (where {{M|\mathcal{H} }} is a [[hereditary sigma-ring]]) that [[the set of all mu*-measurable sets is a ring]]. It is in fact not only a [[ring of
    521 B (82 words) - 01:01, 30 May 2016
  • Notes:The set of all mu*-measurable sets is a ring
    In [[Books:Measure Theory - Paul R. Halmos|Measure Theory]] Halmos does something weird for section 11, theorem B. I have yet to "cra ...mathcal{H} }} and if {{M|\mathcal{S} }} is the set of all [[mu*-measurable set|{{M|\mu^*}}-measurable sets]] then {{M|\mathcal{S} }} is a [[sigma-ring]].
    4 KB (828 words) - 03:11, 30 May 2016
  • Notes:Measures - Real and Abstract Analysis - Hewitt & Stromberg
    ...{M|X}}, a function, {{M|\mu}} defined on {{M|\mathcal{P}(X)}} (the [[power-set]] of {{M|X}}) is a Carathéodory Outer Measure if: {{Notes|Measure Theory}}
    452 B (71 words) - 18:04, 31 May 2016
  • Real numbers
    {{Definition|Real Analysis|Functional Analysis|Set Theory|Analysis|Topology}}
    108 B (13 words) - 21:35, 26 February 2017
  • Group/New page
    Let {{M|G}} be a [[set]] and a [[binary operation]] (a [[function]]) {{M|*:G\times G\rightarrow G} {{Group theory navbox|plain}}
    2 KB (326 words) - 11:38, 2 July 2016
  • Notes:Equivalence relations
    Let {{M|X}} be a set and let {{M|\sim}} be an [[equivalence relation]], then: ...set]] and {{M|\sim}} an [[equivalence relation]], {{M|X/\sim}} denotes the set of [[equivalence class|equivalence classes]] of {{M|\sim}}.
    1 KB (220 words) - 16:56, 12 July 2016
  • Group factorisation theorem
    ...play are eligible (satisfy the requirements to factor) for the theorem. We set up as follows: {{Group theory navbox|plain}}
    7 KB (1,195 words) - 22:55, 3 December 2016
  • Function factorisation theorem
    {{Theorem Of|Set Theory|Abstract Algebra}}
    91 B (12 words) - 18:30, 8 October 2016
  • Logical implication
    {{Definition|Set Theory}}
    47 B (5 words) - 00:06, 16 July 2016
  • Injective
    {{Definition|Set Theory}}
    49 B (5 words) - 17:12, 16 July 2016
  • Equivalence relations
    {{Definition|Set Theory|Abstract Algebra}}
    77 B (8 words) - 04:49, 20 July 2016
  • Free monoid generated by
    ...to check [[Discussion of the free monoid and free semigroup generated by a set]], as there are some things to note Given a [[set]], {{M|X}}, there is a ''free'' [[monoid]], {{M|(F,*)}}{{rAAPAG}}.
    2 KB (419 words) - 16:20, 20 July 2016
  • Free semigroup generated by
    ...nce, see [[Discussion of the free monoid and free semigroup generated by a set]] ...p) - see [[discussion of the free monoid and free semigroup generated by a set]]){{rAAPAG}}, defined as follows:
    1 KB (200 words) - 07:07, 21 July 2016
  • Semigroup
    A semigroup{{rAAPAG}} is a [[tuple]], {{M|(S,*)}}, consisting of a [[set]], {{M|S}} and a [[binary operation]], {{M|*:S\times S\rightarrow S}}, wher {{Semigroup theory navbox|plain}}
    631 B (99 words) - 07:27, 21 July 2016
  • Permutation
    * [[Permutation (combinatorics)]] - a selection of elements from a set where order matters ...et]] - a ''[[bijective]]'' [[mapping]] from a [[set]], {{M|X}} to the same set, {{M|X}}.
    400 B (58 words) - 23:42, 21 July 2016
  • Permutation on a set
    #REDIRECT [[Permutation of a set]] {{Definition|Set Theory|Abstract Algebra|Group Theory}}
    90 B (12 words) - 23:43, 21 July 2016
  • Permutation of a set
    : '''Note: ''' [[permutation on a set]] redirects here. Let {{M|X}} be any ''non-empty'' [[set]], {{M|X}}. A ''permutation'' on {{M|X}}{{rRFAGRBJTA}}{{rAAPAG}} is:
    870 B (134 words) - 23:59, 21 July 2016
  • Notes:The foundations of Mathematics - Kunen
    ==Chapter I: Set Theory== ! Power set
    2 KB (342 words) - 02:38, 31 July 2016
  • Topological group
    ...is a [[Tuple|{{M|3}}-tuple]], {{M|(G,*,\mathcal{J})}} where {{M|G}} is a [[set]], {{M|*:G\times G\rightarrow G}} is a [[binary operation]] (a [[map]] wher * '''Underlying [[set]]: ''' {{M|G}}.
    1 KB (224 words) - 06:23, 8 August 2016
  • Notes:Advanced Linear Algebra - Roman/Chapter 1
    * The set of all finite sums of vectors from the [[union]] {{MM|1=\bigcup_{\alpha\in It is "easy to see" that, under set inclusion as the [[partial ordering]]:
    8 KB (1,463 words) - 14:35, 13 August 2016
  • Extending pre-measures to measures
    # [[The set of all mu*-measurable sets forms a ring|the set of all {{M|\mu^*}}-measurable sets forms a ring]] # [[The set of all mu*-measurable sets forms a sigma-ring|the set of all {{M|\mu^*}}-measurable sets forms a {{sigma|ring}}]]
    2 KB (257 words) - 17:27, 17 August 2016
  • The set of all mu*-measurable sets forms a sigma-ring
    #REDIRECT [[The set of all mu*-measurable sets is a sigma-ring]] {{Theorem Of|Measure Theory}}
    94 B (15 words) - 18:12, 17 August 2016
  • The set of all mu*-measurable sets forms a ring
    #REDIRECT [[The set of all mu*-measurable sets is a ring]] {{Theorem Of|Measure Theory}}
    88 B (15 words) - 18:13, 17 August 2016
  • Semi-ring of sets
    ...is almost a measure. A [[ring of sets]] is closed under all the elementary set operations. ...R} }}, Suppose {{M|a<b}} and {{M|c<d}} (as if either interval is the empty set the result is trivial). Suppose they partially intersect with {{M|a<c}} and
    3 KB (508 words) - 17:25, 18 August 2016
  • The ring of sets generated by a semi-ring is the set containing the semi-ring and all finite disjoint unions
    ...tion</ref> {{M|\mathcal{F} }}, written {{M|R(\mathcal{F})}} is exactly the set {{M|\mathcal{F} }} and all finite [[union|unions]] of elements of {{M|\math ...e proof of this is easy, as [[the intersection of sets is a subset of each set]] we see {{M|1=A\cap B_i\subseteq B_i}} for each {{M|i}}. As the {{M|B_i}}
    7 KB (1,398 words) - 18:33, 19 August 2016
  • Distributivity of intersections across unions
    ...sg=Created for use with [[the ring of sets generated by a semi-ring is the set containing the semi-ring and all finite disjoint unions]], the theorem is e {{Theorem Of|Set Theory|Elementary Set Theory}}
    730 B (128 words) - 23:19, 18 August 2016
  • A pre-measure on a semi-ring may be extended uniquely to a pre-measure on a ring
    * The ring generated by a semi-ring is exactly the set of all finite disjoint unions of elements from that semiring. # [[the ring of sets generated by a semi-ring is the set containing the semi-ring and all finite disjoint unions]]
    2 KB (390 words) - 22:16, 19 August 2016
  • Types of set algebras/Type table
    ...e, if you have come here from a search engine you want to go to [[types of set algebras]]''' ! [[Ring of sets|Ring]]<ref name="PTACC">Probability Theory - A comprehensive course - second edition - Achim Klenke</ref>{{rMTH}}
    4 KB (573 words) - 20:00, 19 August 2016
  • Types of set algebras/Measure theory summary table
    ...from a search engine you don't want this sub-page, but the page [[types of set algebras]] instead''' ==Measure theory perspective summary table==
    559 B (79 words) - 20:05, 19 August 2016
  • Doctrine:Measure theory terminology/Proposals
    This is a sub page for making proposals to the measure theory terminology doctrine. New requests only must be placed here. Queries and su I propose that rather than {{plural|mu*-measurable set|s}} we instead use [[outer splicing sets]] or just [[splicing sets]]. Curre
    3 KB (466 words) - 21:29, 20 August 2016
  • Outer splicing set
    # Unite this with the [[mu*-measurable set]] page, possibly by redirecting it here ...t. It is not a well known term. [[mu*-measurable set|{{M|\mu*}}-measurable set]] redirects here.
    2 KB (378 words) - 22:09, 20 August 2016
  • Hereditary set system
    {{Definition|Set Theory|Measure Theory|Elementary Set Theory}}
    102 B (13 words) - 16:22, 22 August 2016
  • Hereditary set
    An ''hereditary'' set is a set that contains only other sets. ...t - that is to say if this means a set can contain arbitrary sets, or if a set must contain other ''hereditary'' sets}}
    810 B (131 words) - 16:46, 22 August 2016
  • Hereditary system
    {{Definition|Measure Theory|Set Theory|Elementary Set Theory}}
    102 B (13 words) - 14:11, 23 August 2016
  • Hereditary class of sets
    {{Definition|Measure Theory|Set Theory|Elementary Set Theory}}
    102 B (13 words) - 14:11, 23 August 2016
  • Partial-function
    ...">This is my own term. With total orderings any two elements of underlying set of the relation must be comparable. With a total function, {{M|g}}, {{M|g}} ...} (here {{M|f^{-1}(B)}} denotes the [[pre-image]] of {{M|B}}, which is the set containing all {{M|a\in A}} such that {{M|f}} relates {{M|a}} to a {{M|b\in
    2 KB (462 words) - 22:26, 23 August 2016
  • Hereditary system generated by
    ...\in\mathcal{P}(B))]}}</ref> where {{M|\mathcal{P}(A)}} denotes the [[power set]] of {{M|A}}. {{Definition|Measure Theory|Set Theory|Elementary Set Theory}}
    2 KB (371 words) - 23:36, 23 August 2016
  • Sets
    #REDIRECT [[Set]] {{Definition|Set Theory|Elementary Set Theory}}
    65 B (8 words) - 23:45, 23 August 2016
  • Task:Rings and sigma-rings
    # [[the power-set of any set is a ring of sets]] - create # [[the power-set of any set is a sigma-ring of sets]] - create
    509 B (91 words) - 15:52, 30 August 2016
  • First order language
    {{Provisional page|grade=A|msg=Needed for set theory}} ** {{M|V}} - The set of ({{amcm}}, possibly empty) variable symbols: {{M|x_1,x_2,\ldots,x_n,\ldo
    3 KB (455 words) - 10:45, 8 September 2016
  • Domain (FOL)
    ...ty bad that this requires a notion of sets when I want to use this for set theory}} * {{M|M}} is a ''[[non-empty]]'' [[set]] {{Caution|I am studying this for set theory, so something is needed here}}
    4 KB (672 words) - 06:42, 10 September 2016
  • Composition of functions
    {{Definition|Set Theory|Elementary Set Theory}}
    82 B (9 words) - 12:24, 15 September 2016
  • Demonstrating why category arrows are best thought of as arrows and not functions
    ...ry|s}} are ''[[finite]]'' {{plural|set|s}} and whose {{link|arrow|category theory|s}}, {{M|\xymatrix{A \ar[r]^f & B} }} are {{plural|function|s}}{{rAITCTHS20 {{Category theory navbox|plain}}
    2 KB (275 words) - 12:29, 15 September 2016
  • Homotopy is an equivalence relation on the set of all continuous maps between spaces
    ** Let {{M|C^0(X,Y)}} denote the [[set]] of all [[continuous maps]] of the form {{M|(:X\rightarrow Y)}} {{Homotopy theory navbox|plain}}
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  • The relation of maps being homotopic is an equivalence relation
    #REDIRECT [[Homotopy is an equivalence relation on the set of all continuous maps between spaces]] {{Theorem Of|Topology|Homotopy Theory}}
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  • Maps
    {{Definition|Elementary Set Theory}}
    149 B (22 words) - 22:29, 18 September 2016
  • Disjoint union
    ...(set)]] - The {{link|coproduct|category theory}} construction, in the ''[[SET]]'' [[category]] ...cal construct, defining a topology on the disjoint union of the underlying set of a family of topological spaces
    552 B (86 words) - 23:29, 25 September 2016
  • Disjoint union (set)
    ...[sets]]. We denote their ''disjoint union'' or ''{{link|coproduct|category theory}}'' as {{M|1=\coprod_{\alpha\in I}X_\alpha}} and we define this to be: {{Todo|Construction as a set}}
    1 KB (210 words) - 20:21, 25 September 2016
  • Doctrine:Homotopy terminology
    * Let {{M|C^0(X,Y)}} denote [[the set of continuous maps between spaces|the set of continuous maps between {{Top.|X|J}} and {{Top.|Y|K}}]]<ref group="Note" {{Doctrine Of|Topology|Homotopy Theory}}
    3 KB (535 words) - 09:01, 31 October 2016
  • Every injection yields a bijection onto its image
    * {{M|1=f(X):=\{y\in Y\ \vert \exists x\in X[f(x)=y] \} }} - the set of all things in {{M|Y}} that are mapped to by {{M|f}} for some {{M|x\in X} {{Definition|Set Theory|Elementary Set Theory}}[[Category:First-year friendly]]
    4 KB (813 words) - 11:53, 26 September 2016
  • Every bijection yields an inverse function
    {{Definition|Elementary Set Theory|Set Theory}}
    588 B (100 words) - 12:24, 26 September 2016
  • Every surjective map gives rise to an equivalence relation
    {{Theorem Of|Elementary Set Theory|Set Theory}}
    2 KB (315 words) - 13:54, 8 October 2016
  • Disjoint in a set
    #REDIRECT [[Disjoint#Disjoint in a set]] {{Definition|Elementary Set Theory|Set Theory}}
    88 B (12 words) - 03:19, 1 October 2016
  • Disjoint
    ===Disjoint in a set=== Let {{M|Z}} be a set and let {{M|A}} and {{M|B}} be sets (with no other requirements), then we s
    2 KB (294 words) - 03:19, 1 October 2016
  • Non-empty
    A [[set]], {{M|A}} is ''non-empty'' if: ...is non-empty (see "[[disjoint in a set|disjoint in]]" also, "[[Empty in a set|empty in]]" too)
    727 B (124 words) - 04:56, 1 October 2016
  • Non-empty in
    {{Definition|Elementary Set Theory|Set Theory}}
    84 B (10 words) - 04:57, 1 October 2016
  • Disjoint in
    #REDIRECT [[Disjoint#Disjoint in a set]] {{Definition|Set Theory|Elementary Set Theory}}
    88 B (12 words) - 00:35, 2 October 2016
  • Fibre
    * A ''fibre'' of {{M|f}} is any set of the form {{M|f^{-1}(\{y\})}} for some {{M|y\in Y}} * [[Level set]] - a similar concept, rarely used in the same context as a fibre however
    735 B (128 words) - 12:59, 16 October 2016
  • Equivalence relation induced by a function
    {{Definition|Elementary Set Theory|Set Theory}}
    2 KB (276 words) - 22:40, 8 October 2016
  • Canonical projection of an equivalence relation
    Let {{M|X}} be a [[set]] and {{M|\sim\subseteq X\times X}} an [[equivalence relation]] on {{M|X}}. {{Definition|Elementary Set Theory|Set Theory|Abstract Algebra}}
    697 B (107 words) - 22:33, 8 October 2016
  • Canonical projection of the equivalence relation
    {{Definition|Abstract Algebra|Elementary Set Theory|Set Theory}}
    126 B (15 words) - 22:44, 8 October 2016
  • Factoring a function through the projection of an equivalence relation induced by that function yields an injection
    # We must check the set up satisfies the requirements of the {{link|passing-to-the-quotient|functio {{Theorem Of|Elementary Set Theory|Abstract Algebra|Set Theory}}
    6 KB (1,097 words) - 20:24, 9 October 2016
  • Passing-to-the-quotient (function)
    {{Definition|Abstract Algebra|Elementary Set Theory|Set Theory}}
    113 B (14 words) - 18:13, 9 October 2016
  • Exercises:Mond - Topology - 1/Question 8
    Which theorem of [[Group Theory (subject)|group theory]] does this resemble? ...]], then {{M|\frac{X}{\sim} }} is compact also as [[the image of a compact set is compact]] and {{M|\pi:X\rightarrow\frac{X}{\sim} }} is continuous (see [
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  • Surjectivity
    {{Definition|Elementary Set Theory|Set Theory}}
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  • Properties of the pre-image of a map
    {{Theorem Of|Elementary Set Theory|Set Theory}}
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  • Equivalence classes
    {{Definition|Abstract Algebra|Set Theory}}
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  • Ring/New page
    Let {{M|R}} be a ''[[non-empty]]'' [[set]], let there be two [[binary operations]] (a kind of [[map]] where rather t {{Definition|Abstract Algebra|Ring Theory}}
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  • Saturated (map)
    #REDIRECT [[Saturated set with respect to a function]] {{Definition|Elementary Set Theory|Set Theory}}
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  • Saturated
    * [[Saturated set with respect to a function]] - generalisation of fibre {{Definition|Elementary Set Theory|Set Theory}}
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  • Equivalent conditions to a set being saturated with respect to a map
    #REDIRECT [[Equivalent conditions to a set being saturated with respect to a function]] {{Theorem Of|Elementary Set Theory|Set Theory}}
    135 B (20 words) - 13:05, 16 October 2016
  • Saturated set with respect to a function
    * [[Equivalent conditions to a set being saturated with respect to a map]] * [[Fibre]] - this (saturated set) is a generalisation of a fibre.
    785 B (138 words) - 13:08, 16 October 2016
  • Saturated with respect to a function
    #REDIRECT [[Saturated set with respect to a function]] {{Definition|Set Theory|Elementary Set Theory}}
    102 B (14 words) - 13:14, 16 October 2016
  • Fibre (map)
    {{Definition|Elementary Set Theory|Set Theory}}
    67 B (8 words) - 13:14, 16 October 2016
  • Equivalent conditions to a set being saturated with respect to a function
    {{Theorem Of|Elementary Set Theory|Set Theory}}
    807 B (142 words) - 13:15, 16 October 2016
  • Mappings
    {{Definition|Set Theory|Elementary Set Theory}}
    70 B (8 words) - 20:26, 19 October 2016
  • Tuples
    {{Definition|Elementary Set Theory|Set Theory}}
    67 B (8 words) - 20:34, 19 October 2016
  • Direct product module
    * {{M|\prod_{\alpha\in I}M_\alpha}} is the underlying set of the module (we define {{M|1=M:=\prod_{\alpha\in I}M_\alpha}} for conveni * [[Direct sum of modules]] - instances of a {{link|co-product|category theory}}
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  • Subset of
    A set {{M|A}} is a ''subset of'' a set {{M|B}} if {{M|A\subseteq B}}, that is (by the [[implies-subset relation]]) ...n {{M|A\in\mathcal{P}(B)}} (where {{M|\mathcal{P}(B)}} denotes the [[power-set]] of {{M|B}}, by definition, all subsets of {{M|B}}!), or {{M|A\subseteq B}
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  • The intersection of two sets is non-empty if and only if there exists a point in one set that is in the other set
    * Extend that to [[the intersection of sets is a subset of each set]] {{Theorem Of|Elementary Set Theory}}
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  • Intersection is commutative
    {{Theorem Of|Elementary Set Theory|Set Theory}}
    91 B (11 words) - 19:32, 28 October 2016
  • Set-subtraction
    #REDIRECT [[Set subtraction]] {{Definition|Elementary Set Theory|Set Theory}}
    77 B (9 words) - 20:28, 1 November 2016
  • Constant loop based at a point
    #* Let {{M|U\in\mathcal{J} }} be given (so {{M|U}} is an [[open set]] in {{Top.|X|J}}) {{Definition|Topology|Homotopy Theory}}
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  • Constant map
    {{Definition|Set Theory|Elementary Set Theory}}[[Category:Function Terminology]]
    112 B (12 words) - 21:04, 1 November 2016
  • Constant function
    {{Definition|Elementary Set Theory|Set Theory}}[[Category:Function Terminology]]
    977 B (169 words) - 21:12, 1 November 2016
  • Constant mapping
    {{Definition|Elementary Set Theory|Set Theory}}[[Category:Function Terminology]]
    112 B (12 words) - 21:12, 1 November 2016
  • Omega(X,b)
    ...e given. Then {{M|\Omega(X,b)\subseteq}}[[C(I,X)|{{M|C([0,1],X)}}]] is the set containing all [[loops]] based at {{M|b}}{{rITTMJML}}. That is: This set and the operation of [[loop concatenation]] are a precursor for [[the funda
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  • Functions
    {{Definition|Elementary Set Theory|Set Theory}}[[Category:Function Terminology]]
    103 B (11 words) - 05:03, 3 November 2016
  • Proof that the fundamental group is actually a group
    ...{M|b\in X}} be given. Then [[Omega(X,b)|{{M|\Omega(X,b)}}]] denotes the "''set of all [[loops]] based at {{M|b}} in {{M|X}}''"<ref group="Note">Which is a {{Theorem Of|Algebraic Topology|Homotopy Theory|Topology}}
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  • Quotient by the equivalence relation
    {{Definition|Elementary Set Theory|Set Theory}}
    97 B (12 words) - 16:17, 4 November 2016
  • Function factorisation
    {{Theorem Of|Elementary Set Theory|Set Theory}}
    96 B (13 words) - 16:58, 4 November 2016
  • Proof that the fundamental group is actually a group/Outline
    ...note the relation of [[end-point-preserving homotopy]] on {{C(I,X)}} - the set of all {{link|path|topology|s}} in {{M|X}} - but considered only on the sub {{Theorem Of|Homotopy Theory|Topology|Algebraic Topology}}
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  • Homotopy invariance of loop concatenation
    ...ll(0)=\ell(1)=b}}.<br/><br/>Furthermore, {{M|\Omega(X,b)}} is not just a [[set]], it does have a [[group]] structure we can imbue on it, called [[the fund {{Theorem Of|Algebraic Topology|Homotopy Theory|Topology}}
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  • Invariant of an equivalence relation
    ...bra]] where {{M|W}} is quite often used for [[vector spaces]]</ref> be any set and let {{M|f:S\rightarrow W}} be any [[function]] from {{M|S}} to {{M|W}}. * [[Complete system of invariants]] - a finite set of complete invariants really.
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  • Invariant under
    {{Definition|Abstract Algebra|Set Theory|Elementary Set Theory}}
    115 B (14 words) - 18:52, 9 November 2016
  • Completely invariant under
    {{Definition|Elementary Set Theory|Set Theory|Abstract Algebra}}
    134 B (16 words) - 18:54, 9 November 2016
  • Quotient by an equivalence relation
    Let {{M|X}} be a [[set]] and let {{M|\sim\subseteq X\times X}} be an [[equivalence relation]] on { Yes, {{M|\frac{X}{\sim} }} is the set of equivalence classes, it is that simple.
    2 KB (295 words) - 14:16, 13 November 2016
  • Equivalence classes are either equal or disjoint
    ...{{M|\sim}}]]<ref group="Note">In other words: {{M|\frac{X}{\sim} }} is the set of [[equivalence classes]] of {{M|\sim}}</ref> and lastly let {{M|\pi:X\rig {{Theorem Of|Abstract Algebra|Elementary Set Theory|Set Theory}}
    2 KB (388 words) - 14:36, 13 November 2016
  • Polynomial u-ring
    ...be a [[u-ring]] with the standard operations of {{M|+}} and {{M|*}}. The ''set (or further-more: u-ring) of polynomials over {{M|R}}'' (in the ''indetermi ...bject is countable]] and [[finitely many elements removed from a countable set remains countable]]
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  • Absolute value (object)
    ...w\mathbb{R} }} for a [[vector space]] {{M|V}} with almost exactly the same set of properties (3 requires modification) {{Definition|Abstract Algebra|Functional Analysis|Ring Theory|Real Analysis}}
    2 KB (350 words) - 05:25, 21 November 2016
  • Proof that the symmetric group is actually a group
    #REDIRECT [[The collection of all permutations of a set forms a group under function composition]] {{Theorem Of|Group Theory|Abstract Algebra}}
    143 B (21 words) - 11:14, 30 November 2016
  • Symmetric group
    ...b{N} }}. The [[set]] of the [[group]] is the set of all [[Permutation of a set|permutations]] on {{M|\{1,2,\ldots,k-1,k\} }}. See [[proof that the symmetr ...notation quickly becomes heavy so we switch to {{link|cycle notation|group theory}}, which we demonstrate below.
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  • A continuous map induces a homomorphism on fundamental groups
    ...{{M|\mathbb{P}_X^{-1}([f]) }} in the notation developed next, giving us a set of all things equivalent to {{M|f}} and for any of these {{M|\varphi_\ast}} ...ef><ref group="Note">Recall that [[Omega(X,b)|{{M|\Omega(X,p)}}]] is the [[set]] of all {{link|loop|topology|s}} in {{M|X}} based at {{M|p\in X}}. There i
    8 KB (1,475 words) - 07:35, 14 December 2016
  • If the composition of two functions is a bijection then the initial map is injective and the latter map is surjective
    {{Theorem Of|Set Theory|Elementary Set Theory}}
    2 KB (462 words) - 08:27, 14 December 2016
  • The intersection of an arbitrary family of Dynkin systems is itself a Dynkin system
    {{Stub page|grade=A|msg=Needed for Measure Theory ...}_{\alpha\in I} }} be an [[arbitrary family]] of [[Dynkin systems]] on a [[set]] {{M|X}}, then{{rMIAMRLS}} we claim:
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  • Index of notation/L
    | Set of all [[linear maps]], {{M|(:V\rightarrow W)}} - is a [[vector space]] in ! rowspan="2" |{{M|L}}<br/>(Measure Theory<br/>/<br/>Functional Analysis)
    2 KB (260 words) - 21:46, 20 December 2016
  • The reals
    {{Definition|Analysis|Functional Analysis|Metric Space|Set Theory}}
    98 B (12 words) - 21:36, 26 February 2017
  • Relations
    {{Definition|Set Theory}}
    48 B (5 words) - 02:48, 2 August 2017
  • Order terminology
    If {{M|(X,\preceq)}} is a [[poset]], a set with a [[partial order]], then we get a [[strict partial order]], {{M|\prec ...t a total order provided you are looking at sets in the [[power set]] of a set with more than one element). Notice that for subsets of {{M|\{1,2,3\} }} th
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  • Emptyset
    #REDIRECT [[Empty set]] {{Definition|Elementary Set Theory|Set Theory}}
    71 B (9 words) - 11:33, 17 January 2017
  • Empty-set
    #REDIRECT [[Empty set]] {{Definition|Set Theory|Elementary Set Theory}}
    71 B (9 words) - 11:33, 17 January 2017
  • Empty set
    ...iom and is the set that contains no elements. It is also a subset of every set {{Definition|Set Theory|Elementary Set Theory}}
    280 B (46 words) - 11:34, 17 January 2017
  • CW-complex
    # A set {{M|A\in\mathcal{P}(X)}} is [[closed set|closed]] {{iff}} {{M|A\cap\overline{e_\alpha} }} is closed for each {{M|\al {{Definition|Algebraic Topology|Homology|Homotopy Theory|Topology}}
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  • Index of elementary set theory equalities
    [[Category:Index]]{{Theorem Of|Elementary Set Theory|Set Theory}}
    497 B (85 words) - 14:46, 31 January 2017
  • A cap (B-C) = (A cap B) - C
    * [[Index of elementary set theory equalities]] {{Theorem Of|Elementary Set Theory|Set Theory}}
    424 B (71 words) - 15:07, 31 January 2017
  • A-(A-B) = A cap B
    * [[Index of elementary set theory equalities]] {{Theorem Of|Elementary Set Theory|Set Theory}}
    410 B (68 words) - 15:08, 31 January 2017
  • Pre-image (map)
    {{Definition|Elementary Set Theory|Set Theory}}
    82 B (9 words) - 18:31, 31 January 2017
  • Pre-image
    ...Pre-image (function)]] - {{M|f^{-1}(A)}} for a [[function]], {{M|f}} and a set {{M|A}} {{Definition|Elementary Set Theory}}
    228 B (35 words) - 22:28, 31 January 2017
  • Successor function
    #REDIRECT [[Successor of a set]] {{Definition|Set Theory|Advanced Set Theory}}
    78 B (11 words) - 15:10, 3 February 2017
  • Successor of a set
    Let {{M|X}} be a [[set]]. The ''successor'' of the set {{M|X}}, written {{M|S(x)}}, is defined as follows{{rITSTHJ}}: '''Claim 1:''' such a set exists
    2 KB (305 words) - 15:14, 3 February 2017
  • Inductive set
    Let {{M|I}} be a [[set]]. We call {{M|I}} an ''inductive set'' if{{rITSTHJ}} both of the following properties hold: # {{M|\forall n[n\in I\implies}}[[Successor of a set|{{M|S(n)}}]]{{M|\in I]}} - often written as "if {{M|n\in I}} then {{M|(n+1)
    924 B (162 words) - 15:56, 3 February 2017
  • Successor set
    #REDIRECT [[Successor of a set]] {{Definition|Set Theory}}
    58 B (8 words) - 15:46, 3 February 2017
  • Notes:Delta complex/Formal attempt
    * Let {{M|S_n(K)}} be the set of {{n|{{plural|simpl|ex|ices}}}} of {{M|K}} ...ork including this and come up with 2 "null objects" that do not alter the theory, for now {{M|\text{Dim}(\emptyset)\eq -1}} will do. It wont matter.
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  • Trivial
    ...logy induced by a metric|induces]] the [[discrete topology]] - [[the power-set]] topology. ...ract Algebra|Group Theory|Ring Theory|Module Theory|Linear Algebra|Measure Theory|Topology|Metric Space}}
    1 KB (171 words) - 13:29, 16 February 2017
  • Trivial group
    Let {{M|G:\eq\{e\} }}, the set containing one object, which we shall call {{M|e}}, and consider the [[bina {{Definition|Abstract Algebra|Group Theory}}
    2 KB (268 words) - 13:41, 16 February 2017
  • Singleton
    * [[Singleton (set theory)]] - a set with one element, {{M|\{x\} }} is the ''singleton'' containing just {{M|x}} {{Definition|Programming|Set Theory|Elementary Set Theory}}
    360 B (53 words) - 17:53, 17 February 2017
  • Abstract simplicial complex
    ...in\mathbb{N}_0]}} - where {{M|\vert\cdot\vert}} denotes [[cardinality of a set]] ...llow both the empty set to be an ''asc'' and we may also allow the [[empty set]] to be a simplex}} - as per [[Books:Combinatorial Algebraic Topology - Dmi
    3 KB (428 words) - 11:54, 19 February 2017
  • Vertex set (abstract simplicial complex)
    #REDIRECT [[Vertex set of an abstract simplicial complex]] {{Definition|Homology Theory|Simplicial Homology Theory|Algebraic Topology}}
    135 B (16 words) - 11:30, 19 February 2017
  • Vertex scheme of an abstract simplicial complex/Definition
    ...set|simplicial complex}} of {{M|K}} (not to be confused with the [[vertex set of an abstract simplicial complex]]), then we may define {{M|\mathcal{K} }} ...needs to be rewritten!}}</ref> - that is to say {{M|\mathcal{K} }} is the set containing all collections of vertices such that the vertices span a simple
    1 KB (224 words) - 11:51, 19 February 2017
  • Vertex set of an abstract simplicial complex/Definition
    ...mathcal{S} }} be a [[abstract simplicial complex]], we define the ''vertex set'' of {{M|\mathcal{S} }}, denoted as just {{M|V}} or {{M|V_\mathcal{S} }}, a {{Definition|Homology Theory|Simplicial Homology Theory|Algebraic Topology}}
    666 B (102 words) - 11:37, 19 February 2017
  • Evenly covered by a continuous map
    ...inuous map]]. Let {{M|U\in\mathcal{J} }} be given, so {{M|U}} is an [[open set]] of {{Top.|X|J}}, we say that: * Notice that {{M|f^{-1}(U)}} is [[open set|open]] as {{M|f}} is continuous, and that the union of an arbitrary family
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  • Unique lifting property
    ...cted neighbourhoods. It's worth investigating but it isn't critical to the theory.</ref> ...e see that the ''only'' sets that are both [[open sets|open]] and [[closed set|closed]] are {{M|Y}} itself and {{M|\emptyset}}, if the result holds (which
    13 KB (2,510 words) - 16:23, 2 March 2017
  • Simple function under-approximation to a numerical function
    {{Stub page|grade=A|msg=Important for measure theory, and needs a name. SNAF {{M|\leftarrow}} ''simple numerical approximation f ...]}}, then we can construct a (non-negative) {{link|simple function|measure theory}} that (under)-approximates {{M|f}}<ref group="Note">if {{M|a:X\rightarrow\
    2 KB (289 words) - 15:28, 3 March 2017
  • Singleton (set theory)/Definition
    ...">[[Media:WarwickSetTheoryLectureNotes2011.pdf|Warwick lecture notes - Set Theory - 2011 - Adam Epstein]] - page 2.75.</ref>: ...X[t\eq s]}}<ref group="Note">see [[rewriting for-all and exists within set theory]]</ref>
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  • Singleton (set theory)
    Notice that we have manage to define a set containing one thing without any notion of the number 1. #* By "''[[rewriting for-all and exists within set theory]]''" we see:
    1 KB (192 words) - 17:38, 8 March 2017
  • A pair of identical elements is a singleton
    ...{{M|\{t,t\} }}. We now show that {{M|\{t,t\} }} is a {{link|singleton|set theory}}, thus justifying the notation: ...in\{t,t\}\rightarrow y\eq x)]}} (as per definition of {{link|singleton|set theory}}
    2 KB (315 words) - 23:35, 8 March 2017
  • Rewriting for-all and exists within set theory
    {{Theorem Of|Formal Logic|Set Theory}}
    2 KB (358 words) - 17:26, 8 March 2017
  • Axiom schema of replacement
    * Apart of what set theories? ...k|image|function}} of {{M|X}} under {{M|F}} - denoted {{M|F(X)}} is also a set{{rSTTJ}}.
    2 KB (390 words) - 15:28, 5 April 2017
  • Replacement axiom
    {{Definition|Set Theory}}
    67 B (8 words) - 23:18, 8 March 2017
  • Axiom of regularity
    {{Definition|Set Theory}}
    59 B (7 words) - 23:19, 8 March 2017
  • Axiom of foundation
    *** {{ie}} every ''[[non-empty]]'' [[set]] has an [[minimal element|{{M|\in}}-minimal element]] {{Definition|Set Theory}}
    422 B (62 words) - 23:23, 8 March 2017
  • There is no set of all sets
    {{Provisional page|grade=A*|msg=The "core pages" of set theory need to be unified and designed to be viewed in a specific order so all the ...">[[Media:WarwickSetTheoryLectureNotes2011.pdf|Warwick lecture notes - Set Theory - 2011 - Adam Epstein]] - page 6</ref>
    2 KB (330 words) - 01:13, 9 March 2017
  • Metrically bounded set
    ...re other notions of bounded perhaps of sets. As such ''metrically bounded (set)'' ought to be used. This is the start of that process [[User:Alec|Alec]] ( ...name]]" for use with pages</ref> bounded'' or is a ''(metrically) bounded set'' if{{rFAVIDMH}}:
    3 KB (673 words) - 23:10, 18 March 2017
  • A set is bounded if and only if for all points in the space there is a positive real such that the distance from that point to any point in the set is less than the positive real
    ...set being bounded]] which should be moved to [[Equivalent conditions to a set being metrically bounded]] [[User:Alec|Alec]] ([[User talk:Alec|talk]]) 00: ...measured by the [[metric]]) between that first point and any point in the set {{M|A}} is strictly less than {{M|C}}
    6 KB (1,092 words) - 00:41, 19 March 2017
  • Monotonicity of the integral of non-negative extended-real-valued measurable functions with respect to a measure
    ...A*|msg=Stub, needs review, linking to from other pages, a bit of a measure-theory shuffle}} Let {{M|(X,\mathcal{A},\mu)}} be a [[measure space]] and let {{M|f,g\in}}[[Set of non-negative extended-real-valued measurable functions|{{M|\mathcal{M}_{
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  • Simply connected topological space
    ...\ (\text{rel }\{0,1\}))\big]}}]] - note [[C(I,X)|{{M|C([0,1],X)}}]] is the set of all {{link|path|topology|s}} into {{M|X}} {{Definition|Topology|Algebraic Topology|Homotopy Theory}}
    4 KB (601 words) - 16:10, 24 April 2017
  • Contractible topological space
    ...we're okay either way.</ref> - where [[C(X,Y)|{{M|C(X,Y)}}]] is simply the set of all [[continuous maps]] from {{M|X}} to {{M|Y}} {{Definition|Topology|Algebraic Topology|Homotopy Theory}}
    3 KB (544 words) - 20:00, 24 April 2017
  • Homotopy of maps is an equivalence relation
    #REDIRECT [[Homotopy is an equivalence relation on the set of all continuous maps between spaces]] {{Theorem Of|Topology|Homotopy Theory|Algebraic Topology}}
    157 B (22 words) - 20:02, 24 April 2017
  • Homotopy equivalent topological spaces
    ** Here [[C(X,Y)|{{M|C(X,Y)}}]] denotes the set of [[continuous maps]] from {{M|X}} and {{M|f\simeq g}} denotes the relatio {{Definition|Algebraic Topology|Topology|Homotopy Theory}}
    3 KB (596 words) - 21:13, 24 April 2017
  • The diagonal (set)
    ...2-fold [[Cartesian product]] of {{M|X}} with itself, we call the following set the ''diagonal'' of {{M|X}} in {{M|X\times X}}{{rHTJWV}}: {{Definition|Set Theory}}
    802 B (140 words) - 13:58, 12 May 2017
  • Graph
    ...arrow\mathbb{R}^k}} where {{M|U\in\mathcal{P}(\mathbb{R}^n)}} is an [[open set]] and {{M|f}} is [[continuous]] is a [[topological manifold|topological {{n ...Analysis|Manifolds|Topological Manifolds|Topology|Discrete Mathematics|Set Theory}}
    650 B (103 words) - 22:35, 4 June 2017
  • Graph (graph theory)
    * {{M|V}} be a [[finite set]] whose elements we shall call ''vertices'' (singular: ''vertex'') or ''nod * {{M|E}} is a finite set of ''edges'' (singular: ''edge''), which may or may not have data associate
    2 KB (319 words) - 22:15, 13 January 2018
  • Digraph
    ...nguages and automata hardbook by my bed, deal with later.</ref> for finite set of vertices {{M|V}} and {{M|E}} are edges of the form {{M|(v_i,v_j)}} - not {{CS Definition|Graph Theory|Discrete Mathematics}}
    903 B (155 words) - 22:18, 13 January 2018
  • Projection matrix
    ...lgebra, the {{plural|canonical projection|s}} of a {{link|product|category theory}} or a [[projector]] matrix. Otherwise we calculate it's shading (to yield colour and such) and then we set the resulting screen coordinate {{M|x,y}} pixel to whatever colour we want
    6 KB (1,051 words) - 07:44, 13 December 2017
  • Logical workings of the Enigma machines
    ...ments from the [[permutation group]] {{M|S_{26} }}, {{M|I}} represents the set our permutations operate on, which is 26 unique elements (consider these as ...rmutations of the rotors, there may only be 3 rotors, if this is so simply set {{M|D,E:\eq\text{Id} }}
    4 KB (696 words) - 15:24, 15 December 2017
  • Identity map
    ...is if {{M|\text{Id}:X\rightarrow X}} is a [[function]] / [[map]] on some [[set]] {{M|X}}, then: {{Definition|Elementary Set Theory|Set Theory|Abstract Algebra|Category Theory}}
    1 KB (235 words) - 15:06, 15 December 2017
  • Arity
    {{Definition|Set Theory|Elementary Set Theory|Formal Logic}}
    401 B (59 words) - 15:10, 15 December 2017
  • Distribution of the sample median
    ...ution purely on that foundation, then anything which is itself ordinal (in theory both additive and real units) will be a [[corollary]]. ==Set up==
    5 KB (994 words) - 01:22, 16 March 2018
  • Integer division
    {{Definition|Number Theory|Set Theory}}
    304 B (49 words) - 17:06, 7 January 2018
  • Modulo operation
    {{Definition|Number Theory|Analysis|Set Theory}}
    986 B (148 words) - 17:48, 7 January 2018
  • Modulo operator
    {{Definition|Number Theory|Analysis|Set Theory}}
    79 B (9 words) - 17:47, 7 January 2018
  • Restriction (function)
    {{Definition|Set Theory|Elementary Set Theory|Category Theory}}
    89 B (10 words) - 15:35, 12 January 2018
  • Ordered pairs
    {{Definition|Set Theory|Elementary Set Theory|Abstract Algebra}}
    91 B (11 words) - 22:20, 17 January 2018
  • Integers
    ...ion|Abstract Algebra|Number Theory|Real Analysis|Elementary Set Theory|Set Theory}}
    119 B (15 words) - 22:40, 6 February 2018
  • The integers
    ...ion|Abstract Algebra|Number Theory|Real Analysis|Elementary Set Theory|Set Theory}}
    629 B (90 words) - 13:04, 10 February 2018
  • The 2 foundational ways of counting
    ==Relation to set theory== ...he set {{M|A':\eq\{1,\ldots,m\}\subseteq\mathbb{N} }} and with {{M|B}} the set {{M|B':\eq\{1,\ldots,n\}\subseteq\mathbb{N} }} then for:
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  • The 6 primitives of counting
    ...ly some (''suitable'') notion of decision or choice, to then back with set theory, as the caveats below make clear we need [[User:Alec|Alec]] ([[User talk:Al {{Definition|Combinatorics|Set Theory|Elementary Set Theory}}
    720 B (96 words) - 08:46, 31 August 2018
  • Primes (set)
    #REDIRECT [[Set of all prime numbers]] {{Definition|Number Theory}}
    67 B (9 words) - 22:40, 30 September 2018
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