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• Injection
  {{Requires work|grade=A* For a [[Function|function]] <math>f:X\rightarrow Y</math> every element of <math>
  3 KB (463 words) - 21:50, 8 May 2018
• Open ball
  Given a [[metric space]] {{M|(X,d)}} the ''open ball centred at {{M|x_0\in X}} of r ...{x\in X\vert\ d(x,x_0)<r\} }} - that is all the points of {{M|X}} that are a distance (given by {{M|d}}) strictly less than {{M|r}} from {{M|x_0}}
  4 KB (842 words) - 02:00, 29 November 2015
• Continuous map
  {{Refactor notice|grade=A}} ...are a few different conditions for continuity, there's also continuity at a point. This diagram is supposed to show how they relate to each other.
  6 KB (972 words) - 01:44, 14 October 2016
• Surjection
  {{Requires work|grade=A*|msg=See [[Injection]]'s requires-work box [https://wiki.unifiedmathematics ...element of <math>B</math> is mapped onto from at least one thing in <math>A</math>
  2 KB (263 words) - 21:56, 8 May 2018
• The relationship between logical implication and the subset relation
  ...f B") if and only if every element of <math>A</math> also belongs to <math>B</math> That is: <math>[A\subseteq B]\iff\forall x[x\in A\implies x\in B]</math><ref>Definition 3.10 (p10) - Introduction to Set Theory, Third Editi
  776 B (136 words) - 17:36, 16 January 2017
• Basis for a topology
  {{Refactor notice|grade=A|msg=Updating with findings. * Need to add: [[A function is continuous if and only if the pre-image of every basis element
  5 KB (802 words) - 18:35, 17 December 2016
• Homomorphism
  {{Refactor notice|grade=B}} ** [[Homomorphism (vector space)]] - {{AKA}}: [[linear map]] - instance of a [[module homomorphism]]
  4 KB (532 words) - 22:04, 19 October 2016
• The set of all open balls of a metric space are able to generate a topology and are a basis for that topology
  ...n Apr 2015 when [[Template:Theorem]] was moved to [[Template:Theorem Of]]. A very old page indeed!}} ...M|X}} be a [[set]], let {{M|d:X\times X\rightarrow\mathbb{R}_{\ge 0} }} be a [[metric]] on that set and let {{M|(X,d)}} be the resulting [[metric space]
  4 KB (814 words) - 22:16, 16 January 2017
• Triangle inequality
  ...[Metric space|metric space]]) of which <math>|x-z|\le|x-y|+|y-z|</math> is a special case. ...is <math>|a+b|\le |a|+|b|</math>, notice if we set {{M|1=a=x-y}} and {{M|1=b=y-z}} then we get <math>|x-y+y-z|\le|x-y|+|y-z|</math> which is just <math>
  3 KB (546 words) - 13:05, 19 February 2016
• Product topology
  {{Refactor notice|grade=A|As a part of the topology patrol}} {{Requires references|grade=A|msg=Check Munkres and Topological Manifolds}}
  5 KB (871 words) - 20:32, 23 September 2016
• Function
  A function {{M|f}} is a special kind of [[Relation|relation]] ...]<ref name="API">Analysis - Part 1: Elements - Krzysztof Maurin</ref>, for a relation:
  4 KB (659 words) - 13:01, 19 February 2016
• Set theory axioms
  ...posed to provide some discussion for the axioms (for example "there exists a set with no elements" doesn't really deserve its own page) ...eory axioms|motivation for set theory axioms page]], note that "R" denotes a result. Only "major" results are shown, they are covered in the motivation
  3 KB (619 words) - 10:25, 11 March 2015
• Motivation for set theory axioms
  This assumes you have some "general idea" of what a set is. For example you are happy that {{M|\{1,2,3\}=\{2,1,3\} }} and that ...can say "from these axioms, we know this set exists" - for example we use a letter (<math>\emptyset</math>) to denote the empty set, what if there are
  3 KB (584 words) - 23:03, 28 February 2015
• Implies
  ...th>A\implies B</math> or <math>B\impliedby A</math> where {{M|A\text{ and }B}} are statements of some sort means<ref>https://en.wikipedia.org/w/index.ph ! A
  681 B (111 words) - 13:36, 16 June 2015
• Editing guide
  | rowspan="2" | A shame, because a black-board style {{M|1}} would be a great [[indicator function]] ...ome letters can be difficult to identify, for example {{M|\mathscr{I} }} - a {{C|J}} right? ''Wrong'' - an {{C|I}}}}
  5 KB (759 words) - 18:48, 24 September 2016
• Ordered pair
  ...dered pair <math>(a,b)=\{\{a\},\{a,b\}\}</math>, this way <math>(a,b)\ne(b,a)</math>. Suppose we are given <math>a,b</math> (so we can be sure they exist).
  2 KB (327 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) ...oof 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
  ...g|Proof that <math>\le</math> is a partial ordering <math>\iff <</math> is a strict ordering]] ...special kind of [[Relation|relation]], we can define an order uniquely as a partial or strict ordering. That is the two are equivalent.
  5 KB (1,006 words) - 13:21, 1 January 2016
• Equivalence relation
  A [[relation]], {{M|\sim}}, in {{M|X}}<ref group="Note">This terminology mean ...ss]] is the name given to the set of all things which are equivalent under a given equivalence relation.
  3 KB (522 words) - 15:18, 12 February 2019
• Linear map
  | What we'd call linear map, it's just a map {{M|T:(V,F)\rightarrow(W,F)}} where {{M|1=T(\alpha u+\beta v)=\alpha T( | A linear transform into the same space as the domain, that is {{M|T:(V,F)\rig
  3 KB (512 words) - 16:30, 23 August 2015
• Bilinear map
  ...inear map|linear map]] which takes a point in a vector space to a point in a different vector space) ...inear map, and an ''[[Inner product|inner product]]'' is a special case of a bilinear form.
  4 KB (682 words) - 15:44, 16 June 2015
• Norm
  ...ote">The other mistake books make is saying explicitly that the [[field of a vector space]] needs to be {{M|\mathbb{R} }}, it may commonly be {{M|\mathb # <math>\forall x,y\in V\ \|x+y\|\le\|x\|+\|y\|</math> - a form of the [[Triangle inequality|triangle inequality]]
  6 KB (1,026 words) - 20:33, 9 April 2017
• Cauchy-Schwarz inequality
  ...ighlight|Update: [[Cauchy-Schwarz inequality for inner product spaces]] is a proof of the second form - note that {{M|\Vert x\Vert:\eq\sqrt{\langle x,x\ ...he solutions (to <math>f(x)=0</math>) will at be: <math>x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}</math>
  3 KB (609 words) - 13:04, 4 April 2017
• Index of notation
  ...,\mathbb{R})</math> denotes the continuous function on the interval {{M|[a,b]}} that map to {{M|\mathbb{R} }} - this is unlikely to be given any other w | {{M|\mathbb{S}^n}}, {{M|l_2}}, {{M|\mathcal{C}[a,b]}}
  9 KB (1,490 words) - 06:13, 1 January 2017
• Span, linear independence, linear dependence, basis and dimension
  Given a set of vectors {{M|S}} in a vector space {{M|(V,F)}} A set {{M|E}} in a [[Vector space|vector space]] {{M|(V,F)}} is linearly dependent if for any
  2 KB (330 words) - 18:07, 25 April 2015
• Sequence
  A sequence is one of the earliest and easiest definitions encountered, but I ...prefer it. This notation is inline with that of a [[Tuple|tuple]] which is a generalisation of [[Ordered pair|an ordered pair]].
  2 KB (419 words) - 18:12, 13 March 2016
• Basis and coordinates
  This is very much a "motivation" page and a discussion of the topic. ==What is a coordinate==
  9 KB (1,525 words) - 16:30, 23 August 2015
• Formal logic language
  ...y so in set theory, but overall an important role. It is important to have a concrete understanding of this. *<math>\text{Plus}(\text{Natural }a,\text{Natural }b)\rightarrow\text{Natural}</math>
  2 KB (410 words) - 16:35, 9 March 2015
• Group
  {{Requires references|The bulk of this page was written when this was a 'note project' and was taken from books (even though I was already really f ...denotes the group's operation applied to the elements {{M|a\in G}} and {{M|b\in G}}.
  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
• Algebra of sets
  : '''Note: ''' Every ''algebra of sets'' is a ''[[ring of sets]]'' (see below) An ''algebra of sets'' is a collection of sets, {{M|\mathcal{A} }} such that{{rMTH}}:
  3 KB (507 words) - 18:43, 1 April 2016
• Sigma-ring
  A '''Sigma-ring''' or <math>\sigma</math>-ring, is closely related to [[Ring A non-empty class of sets {{M|S}} is a {{sigma|ring}} if<ref>Measure Theory, p24 - Halmos - Graduate Texts in Math
  728 B (125 words) - 15:34, 13 March 2015
• Sigma-algebra
  ...: ''' A ''Sigma-algebra'' of sets, or {{sigma|algebra}} is very similar to a [[Sigma-ring|{{sigma|ring}}]] of sets. ...ts]] is to an [[algebra of sets]] as a [[sigma-ring|{{sigma|ring}}]] is to a ''{{sigma|algebra}}''
  8 KB (1,306 words) - 01:49, 19 March 2016
• Measure Theory
  <math>[[a,b))\in\mathcal{J}^n</math> means <math>[a_1,b_1)\times[a_2,b_2)\times\cdots\t We can clearly get a ring from this, but not a [[Sigma-ring|{{Sigma|ring}}]] as for example:
  4 KB (733 words) - 01:41, 28 March 2015
• Additive function
  ...function]] on a class of [[set|sets]], {{M|\mathcal{A} }}, {{M|f:\mathcal{A}\rightarrow\mathbb{R} }} is called ''additive'' or ''finitely additive'' if ...|1=A\cap B=\emptyset}} ([[pairwise disjoint]]) and {{M|A\udot B\in\mathcal{A} }} we have:
  6 KB (971 words) - 18:16, 20 March 2016
• Pre-measure
  ...ef> is a [[Measure|measure]] on an [[Algebra of sets|algebra]] rather than a [[Sigma-algebra|{{Sigma|algebra}}]], the properties are as follows: ...an algebra of sets (a system of subsets of {{M|X}}) and {{M|\mu_0:\mathcal{A}\rightarrow[0,+\infty]}} such that:
  5 KB (782 words) - 01:49, 26 July 2015
• Measure
  {{Stub page|Requires further expansion|grade=A}}{{Extra Maths}}{{:Measure/Infobox}} A (positive) ''measure'', {{M|\mu}} is a [[set function]] from a [[sigma-ring|{{sigma|ring}}]], {{M|\mathcal{R} }}, to the positive [[extend
  6 KB (941 words) - 14:39, 16 August 2016
• Lebesgue measure
  ...Rene L. Schilling</ref> that assigns every half-open rectangle <math>[\![a,b)\!)=[a_1,b_1)\times\cdots\times[a_n,b_n)\in\mathcal{J}</math> as follows: <math>\lambda^n\big([\![a,b)\!)\big)=\prod^n_{i=1}(b_i-a_i)</math>
  804 B (129 words) - 00:28, 20 December 2016
• Complement
  ...|X}} the complement of {{M|A}} (often denoted {{M|A^c}}, {{M|A'}} or {{M|C(A)}}) is given by: <math>A^c=\{x\in X|x\notin A\}=X-A</math>
  726 B (145 words) - 13:28, 18 March 2015
• Ring
  {{Refactor notice|grade=A|msg=Page was ancient, mostly written in May 2015}} Not to be confused with [[Ring of sets|rings of sets]] which are a topic of [[Algebra of sets|algebras of sets]] and thus [[Sigma-algebra|{{Si
  7 KB (1,248 words) - 05:02, 16 October 2016
• Measurable map
  '''Note: ''' Sometimes called a ''measurable [[Function|fuction]]''<ref name="PAS">Probability and Stochast ...and {{M|(X',\mathcal{A}')}} be [[Measurable space|measurable spaces]] then a map:
  5 KB (792 words) - 02:31, 3 August 2015
• Random variable
  A '''Random variable''' is a [[Measurable map|measurable map]] from a [[Probability space|probability space]] to any [[Measurable space|measurabl ...space]] and let {{M|X:(\Omega,\mathcal{A})\rightarrow(V,\mathcal{U}) }} be a random variable
  4 KB (596 words) - 14:41, 20 March 2015
• Curve
  A curve can mean many things. It is reasonably standard to say however that a curve is any one dimensional "thing" Given a <math>f:\mathbb{R}^n\rightarrow\mathbb{R}</math> and a {{M|c\in\mathbb{R} }} we define the level curve as follows<ref>
  1 KB (224 words) - 21:30, 28 March 2015
• Parametrisation
  A parametrisation {{M|\gamma}} is a function<ref>Elementary Differential Geometry - Pressley - Springer SUMS</r <math>\gamma:(a,b)\rightarrow\mathbb{R}^n</math> with <math>-\infty\le a< b\le +\infty</math>
  1 KB (190 words) - 11:10, 12 June 2015
• Vector dot product
  Given two vectors in a finite vector space {{M|a,b\in V}} where {{M|v_i}} denotes the {{M|i^\text{th} }} element of {{M|v}}, t <math>a\cdot b=\sum^n_{i=1}a_ib_i</math>
  534 B (96 words) - 02:04, 29 March 2015
• Reparametrisation
  This page requires knowledge of a [[Parametrisation|parametrisation]] of a [[Curve|curve]] ...parametrisation'' of the [[Parametrisation|parametrisation <math>\gamma:(a,b)\rightarrow\mathbb{R}^n</math>]] if there exists:
  728 B (115 words) - 16:30, 23 August 2015
• Smooth manifold
  ...opological manifold|topological {{n|manifold}}]] and {{M|\mathcal{A} }} is a [[Smooth structure|smooth structure]] on {{M|M}} ...between their domains. That is given two charts {{M|(A,\alpha)}} and {{M|(B,\beta)}} that:
  3 KB (413 words) - 21:09, 12 April 2015
• Motivation for quotient topology
  ==Convincing yourself the torus is a quotient map== ...off the left side, you appear at the right, this is a torus. If you're on a doughnut and go "up" you'll go around and end up (eventually) back where yo
  4 KB (681 words) - 10:33, 7 April 2015
• Polar equation of a line
  *** {{M|1=a=m^2+1}} *** {{M|1=b=2m}}
  1 KB (223 words) - 22:43, 10 April 2015
• Smooth map
  ...ooth manifold|smooth manifolds]] {{M|(M,\mathcal{A})}} and {{M|(N,\mathcal{B})}} (of not necessarily the same dimension) is said to be smooth<ref>Introd ...\exists\ (U,\varphi)\in\mathcal{A},\ p\in U\text{ and }(V,\psi)\in\mathcal{B}</math> such that <math>F(U)\subseteq V\wedge[\psi\circ F\circ\varphi^{-1}:
  1 KB (235 words) - 21:37, 14 April 2015
• The fundamental group
  {{Refactor notice|grade=A|msg=I cannot believe it's been 15 months and this still isn't complete! ...rel }\{0,1\})\big)}}]] on {{M|C(I,X)}} and restricted to {{M|\text{Loop}(X,b)}}, then:
  3 KB (393 words) - 16:10, 4 November 2016
• Circle
  A circle is usually defined by {{M|1=\mathcal{S}^1=\Big\{(x,y)\in\mathbb{R}^2 ...{M|f:t\mapsto e^{2\pi jt} }} is significant. As it makes {{M|\mathbb{R} }} a [[Covering space|covering space]] of {{M|\mathbb{S}^1}}
  3 KB (592 words) - 16:57, 11 May 2015
• Paths and loops in a topological space
  ==Path in a topological space== A path in {{M|X}} is any [[Continuous map|continuous map]] {{M|p:[0,1]\righta
  2 KB (347 words) - 19:36, 16 April 2015
• Homotopy class
  ...s from {{M|a}} to {{M|b}} (which are not necessarily distinct as it may be a loop) is an equivalence relation, which is to say: * For a [[Continuous map|continuous map]] {{M|p:[0,1]\rightarrow[0,1]}} with {{M|1=
  2 KB (316 words) - 02:37, 17 April 2015
• Hausdorff space
  ...ce|review=true|msg=Page was 1 year and 1 day since modification, basically a stub, seriously needs an update. ...ne with the finite complement topology is not Hausdorff]] as an example of a familiar set with an unfamiliar topology
  4 KB (679 words) - 22:52, 22 February 2017
• Inner product
  ...ame="FA">Functional Analysis - George Bachman and Lawrence Narici</ref> is a map: Given a vector space {{M|X}} over either {{M|\mathbb{R} }} or {{M|\mathbb{C} }}, an
  6 KB (1,016 words) - 12:57, 19 February 2016
• Xypic
  <m> \xymatrix{ %Our diagram is a 4x10 matrix &0\ar[rr]|(.475)\hole&&A'_2\ar[rr]|\hole&&B'_2\ar[rr]|\hole&&C'_2&&\\
  2 KB (325 words) - 17:14, 23 October 2016
• Generated subgroup
  A [[Cyclic subgroup|cyclic subgroup]] is a group generated by a single element. Let {{M|(G,\times)}} be a group, and {{M|\{g_1,\cdots,g_n\}\subset G}} be a set of elements of {{M|G}}, then the ''subgroup generated by {{M|1=\{g_i\}_
  2 KB (404 words) - 12:39, 12 May 2015
• Exercises:Serge Lang - Undergraduate Algebra - C2S1
  ...d {{M|a,\ b,\ c}} be elements of {{M|G}}. If {{M|1=ab=ac}} show that {{M|1=b=c}}
  322 B (54 words) - 00:52, 27 February 2016
• Examples of rings
  Rings here refers to [[Ring|rings]] from abstract algebra, not a [[Ring of sets|ring of sets]] ...addition and multiplication (usual meaning 5+4 = 9, 5*4=20 - so forth) is a:
  2 KB (269 words) - 17:11, 19 May 2015
• Division algorithm
  {{Requires work|grade=A*|msg=Tasks: * Add algorithm - it's the crappy one that just subtracts until {{M|0\le r < b}} actually holds, and adds one to {{M|q}} each iteration [[User:Alec|Alec]]
  3 KB (594 words) - 16:48, 7 December 2017
• Greatest common divisor
  '''Note:''' requires knowledge of what it means for a number to be a [[Divisor|divisor]] of another. ...t positive integer, {{M|d}}, that [[Divisor|divides]] both {{M|a}} and {{M|b}}. We write:
  1 KB (252 words) - 08:33, 21 May 2015
• Divisor
  ...Number theory and RSA cryptography - S. C. Coutinho</ref> if there exists a third integer, {{M|c\in\mathbb{Z} }} such that: * <math>a=bc</math>
  783 B (141 words) - 08:20, 21 May 2015
• Euclidean algorithm
  ...M|a,b\in\mathbb{N}_+}}, such that {{M|a\ge b}} (swap them around if {{M|a< b}}) | The [[GCD]] of {{M|a}} and {{M|b}}
  1 KB (187 words) - 08:53, 21 May 2015
• Passing to the quotient (function)
  : See [[Passing to the quotient]] for a disambiguation of this term. {{float-right|{{/Diagram}}|style=max-width:20em;}}Given a function, {{M|f:X\rightarrow Y}} and another function, {{M|w:X\rightarrow W
  8 KB (1,644 words) - 20:49, 11 October 2016
• Notes:ToMond
  I have phrased this a bit weirdly but that's to make it parallel to tensors, I also use {{M|W}} t ...ear. Here {{M|W}} is "whatever" and is a vector space. {{M|\tilde{f} }} is a linear map
  5 KB (921 words) - 05:43, 7 June 2015
• Integral domain
  ...an update to the modern style of the site, it was also created largely as a stub page}} Given a [[Ring|ring]] {{M|(D,+,\times)}}, it is called an ''integral domain''<ref n
  2 KB (327 words) - 11:09, 20 February 2016
• Ordered integral domain
  ...bstract Algebra - An Expanded Version - Neal H. McCoy</ref> if it contains a subset, which we'll denote {{M|D^+}} with the following properties: # {{M|a,b\in D^+\implies a+b\in D^+}} (closed under addition)
  1,016 B (168 words) - 16:10, 23 August 2015
• Index of norms and absolute values
  | Denotes the [[Norm]] of a vector ...<math>C([a,b],\mathbb{R})</math>, given by <math>\|f\|_\infty=\sup_{x\in[a,b]}(|f(x)|)</math>
  1 KB (207 words) - 09:16, 9 June 2015
• Semi-ring of half-closed-half-open intervals
  ...eq|a_i|i|1|n|in=\mathbb{R}|pre=a:=}}<ref group="Note">Or equivalently, {{M|a\in\mathbb{R}^n}}, either way we get an {{n|[[tuple]]}} of real numbers</ref ...the same length (namely {{M|n\in\mathbb{N} }}), we define {{M|1=\[a,b\)}}, a ''half-open-half-closed rectangle in {{M|\mathbb{R}^n}}''{{rMIAMRLS}} as fo
  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> ...rite this</ref>, this means <math>\forall A,B\in\mathcal{A}[A-B\in\mathcal{A}]</math>
  2 KB (360 words) - 20:43, 15 June 2015
• Properties of classes of sets closed under set-subtraction
  {{Refactor notice|grade=A|msg=The old page was crap, it is preserved below but it doesn't provide any Suppose {{M|\mathcal{A} }} is an arbitrary class of [[set|sets]] with the property that:
  3 KB (490 words) - 11:38, 21 August 2016
• Contrapositive
  .../w/index.php?title=Contraposition&oldid=673682637</ref>: {{M|¬B\implies ¬A}} ! {{M|A}}
  627 B (98 words) - 18:44, 28 August 2015
• Notes:Just what is in a generated sigma-algebra
  Consider a [[Topological space|topology]] on {{M|X}}, call it {{M|\mathcal{J} }}, what ...sigma(\mathcal{J})}} only contains things we can get to via the operations a [[Sigma-algebra|{{Sigma|algebra}}]] allows starting from the open sets, {{M
  1 KB (256 words) - 13:29, 17 June 2015
• Partition (abstract algebra)
  ...rary [[Indexing set|indexing set]], to each element {{M|i\in I}} we assign a set {{M|A_i}} which is non-empty. If: * {{M|1=B=\bigcup_{i\in I}A_i}}
  2 KB (316 words) - 16:03, 18 June 2015
• Site projects:Patrolling measure theory/Task list
  * [[Integral of a simple function (measure theory)]] '''DONE''' [[User:Alec|Alec]] ([[User ta * [[Integral of a simple function (measure theory)/Definition]] '''DONE''' [[User:Alec|Alec]]
  5 KB (645 words) - 11:40, 21 August 2016
• Trinary index
  A trinary index is a way of indexing lists of theorems by their properties. It works as follows: ! class="unsortable" | {{M|A}}
  964 B (165 words) - 20:55, 22 June 2015
• Bounded set
  {{Stub page|grade=A|msg=Needs fleshing out}} {{Refactor notice|grade=A|msg=I've found a definition!}}
  2 KB (409 words) - 23:31, 29 October 2016
• Complete metric space
  ...uence]] converges to a [[limit (sequence)]] within {{M|X}} then {{M|X}} is a ''complete metric space''<ref name="FA">Functional Analysis - George Bachma ...{{M|1=(x_n)_{n=1}^\infty}}, it converging to a limit {{M|x\in X}} or being a [[Cauchy sequence]] are equivalent. Or in symbols:
  2 KB (382 words) - 15:36, 24 November 2015
• Inner product examples
  ...]}} - the [[Continuous map|continuous functions]] over the interval {{M|[a,b]}} that are complex valued. : (this is simpler then it sounds as for {{M|f\in\mathcal{C}_\mathbb{C}[a,b]}} we really have {{M|1=f(x)=f_r(x)+jf_i(x)}} where {{M|1=j:=\sqrt{-1} }})
  3 KB (678 words) - 16:16, 11 July 2015
• Trace sigma-algebra
  {{Stub page|grade=B|More results would be good. Relation to pullback too}} ...|algebra}} on {{M|Y}} called the ''trace {{sigma|algebra}}'', {{M|\mathcal{A}_Y}} given by{{rMIAMRLS}}:
  921 B (144 words) - 12:00, 23 August 2018
• Well-ordered set
  '''Note:''' This page exists only to contain a simpler, easier view of [[Well-ordering]] - until all the concepts can be u ...ing]] {{M|<\subseteq A\times A}} where if {{M|1=(a,b)\in<}} we write {{M|a<b}} is said to be ''well ordered''<ref name="top">Topology - James R. Munkres
  916 B (164 words) - 17:49, 24 July 2015
• Pre-measure/Properties in common with measure
  ...''' if {{M|1=A\cap B=\emptyset}} then {{M|1=\mu_0(A\udot B)=\mu_0(A)+\mu_0(B)}} ...tingales'' in fact!)</ref> if {{M|A\subseteq B}} then {{M|\mu_0(A)\le\mu_0(B)}}
  1 KB (201 words) - 22:30, 30 March 2016
• A function is a measure iff it measures the empty set as 0, disjoint sets add, and it is continuous from below (with equiv. conditions)
  ...ths}}Let {{M|(X,\mathcal{A})}} be a [[Measurable space|measurable space]]. A map: * {{M|\mu:\mathcal{A}\rightarrow[0,\infty]}}
  2 KB (376 words) - 12:12, 27 July 2015
• Discrete metric and topology/Summary
  ! [[Topology induced by a metric|induced topology]] ...M|A}} contains an ''[[Open ball|open ball]]'' centred at each point in {{M|A}}.
  1 KB (263 words) - 13:04, 27 July 2015
• Dynkin system/Definition 2
  ...f subsets of {{M|X}} we denote {{M|\mathcal{D}\subseteq\mathcal{P}(X)}} is a ''Dynkin system''<ref name="PAS">Probability and Stochastics - Erhan Cinlar * {{M|1=\forall A,B\in\mathcal{D}[B\subseteq A\implies A-B\in\mathcal{D}]}}
  719 B (120 words) - 23:22, 2 August 2015
• A map from two sigma-algebras, A and B, is measurable if and only if for some generator of B (call it G) we have the inverse image of S is in A for every S in G
  ...|\mathcal{F} }} we have {{M|\forall S\in\mathcal{F}_0[f^{-1}(S)\in\mathcal{A}]}}}} ...F,\mathcal{F})}}, {{M|f:A\rightarrow F}}, is [[measurable map|{{M|\mathcal{A}/\mathcal{F} }} measurable]] {{iff}} for some [[Generator (sigma-algebra)|g
  2 KB (328 words) - 13:23, 18 March 2016
• Conditions for a map to be a measurable map
  ...ded to be a list of different conditions for a [[Function|function]] to be a [[Measurable map|measurable map]] ...|\mathcal{F} }} we have {{M|\forall S\in\mathcal{F}_0[f^{-1}(S)\in\mathcal{A}]}}]]<ref name="PAS">Probability and Stochastics - Erhan Cinlar</ref>
  678 B (120 words) - 23:54, 2 August 2015
• Composition of measurable maps is measurable
  ...urable (same as saying: {{M|f:A\rightarrow B}} is {{M|\mathcal{A}/\mathcal{B} }}-measurable) and * {{M|g:(B,\mathcal{B})\rightarrow(C,\mathcal{C})}} is measurable
  830 B (129 words) - 00:04, 3 August 2015
• Function terminology
  For a function {{M|f:A\rightarrow B}} where {{M|A}} and {{M|B}} are assumed to be any set unless stated otherwise we say the following ab ! {{M|A}}'s structure
  1 KB (165 words) - 02:07, 3 August 2015
• Min/max
  ...sh out this page - I created it only to document the {{M|1=a\vee b:=\max(a,b)}} seen in Probability and Stochastics}} ! {{M|a\wedge b}}
  884 B (144 words) - 02:51, 3 August 2015
• Borel sigma-algebra generated by
  ...denote the set of half-open-half-closed rectangles in {{M|\mathbb{R}^n}} - a totally separate thing</ref> is any [[Topological space|topology]]) ...- if the topology on {{M|X}} is obvious, we may simply write: {{M|\mathcal{B}(X)}}<ref name="MIM"/>
  2 KB (244 words) - 08:30, 6 August 2015
• Borel sigma-algebra
  ...M|1=\mathcal{B}(X,\mathcal{J}):=\sigma(\mathcal{O})}} or just {{M|\mathcal{B}(X)}} if the topology is implicit. ...hcal{B}(\mathbb{R})}}, with it's usual topology (the [[Topology induced by a metric|topology induced by]] the [[Absolute value|absolute value metric]],
  5 KB (854 words) - 09:25, 6 August 2015
• Compactness/Uniting covers proof
  *A subset {{M|Y\subseteq X}} of a [[Topological space|topological space]] {{M|(X,\mathcal{J})}} is [[Compactn * Every [[Covering|cover]] by sets open in {{M|X}} has a finite subcover. }}
  7 KB (1,411 words) - 19:44, 15 August 2015
• Outer-measure/Definition
  </noinclude>An ''outer-measure'', {{M|\mu^*}} is a [[set function]] from a [[hereditary sigma-ring|hereditary {{sigma|ring}}]], {{M|\mathcal{H} }}, to * {{M|\forall A\in\mathcal{H}[\mu^*(A)\ge 0]}} - non-negative
  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
  ...The proof needs to be checked before this page may be considered no longer a stub}} ...of a set is no more than the sum of the (pre-)measures of the elements of a covering for that set/Statement|Statement]]==
  4 KB (688 words) - 21:03, 31 July 2016
• Dynkin system/Proof that definitions 1 and 2 are equivalent
  : Let {{M|\mathcal{D} }} be a subgroup satisfying definition 1, then I claim it satisfies definition 2. L :# For {{M|A,B\in\mathcal{D} }} with {{M|B\subseteq A}} then {{M|A-B\in\mathcal{D} }}
  2 KB (326 words) - 05:09, 22 August 2015
• Set subtraction
  Given two sets, {{M|A}} and {{M|B}} we define ''set subtraction'' ({{AKA}}: ''relative complement''{{rMTH}}) * {{M|1=A-B=\{x\in A\vert x\notin B\} }}
  1 KB (237 words) - 00:48, 21 March 2016
• Notes:Pick function
  {{Note|This page was written to respond to a [http://gamedev.stackexchange.com/questions/108348/project-2d-click-touch-o ...{{M|4\times 4}} (projection AND view operator) you are using (that is for a point {{M|x}}, {{M|Px}} is the complete transformation to clip coordinates)
  4 KB (686 words) - 01:43, 15 September 2015
• Notes:Spherical coordinates
  ...8667/how-to-load-a-spherical-planet-and-its-regions this question] and not a part of the site, I needed math-markup}} ...on) - it's actually (topologically) a sphere, it's a bit bumpy but mainly a sphere, yet to someone standing on it it is rather plane-like. This was act
  10 KB (1,899 words) - 18:48, 23 September 2015