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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