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• 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
• 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
• 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
• 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
• 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
• Algebra of sets
  * [[Types of set algebras]] {{Measure theory navbox|plain}}
  3 KB (507 words) - 18:43, 1 April 2016
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
  2 KB (217 words) - 15:26, 26 February 2016
• 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
  2 KB (304 words) - 17:01, 20 February 2016
• 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
• 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
• 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
• 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
• 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
  {{Function terminology navbox|plain}} {{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 {{Function terminology navbox|plain}}
  652 B (107 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
• Greater than or equal to
  {{Order theory navbox|plain}} {{Relations navbox}}
  1 KB (152 words) - 15:56, 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
• 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
• 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
• 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
• 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
• 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
• 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 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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}}
  2 KB (272 words) - 23:37, 14 October 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
• 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} {{Function terminology navbox|plain}}
  4 KB (813 words) - 11:53, 26 September 2016
• Every bijection yields an inverse function
  {{Function terminology navbox|plain}} {{Definition|Elementary Set Theory|Set Theory}}
  588 B (100 words) - 12:24, 26 September 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
• 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
  {{Relations navbox|plain}} {{Definition|Elementary Set Theory|Set Theory}}
  2 KB (276 words) - 22:40, 8 October 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.
  3 KB (425 words) - 12:21, 30 November 2016