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- Let {{M|X}} and {{M|Y}} be [[sets]] and let {{M|f:X\rightarrow Y}} be a [[function]] between them. Then:2 KB (417 words) - 00:51, 14 October 2016
- * by [[properties of the pre-image of a function]], the pre-image of a union is the union of pre-images. *** But by [[properties of the pre-image of a function]] we see:4 KB (839 words) - 18:35, 17 December 2016
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- An injective function is 1:1, but not nessasarally [[Surjection|onto]]. For a [[Function|function]] <math>f:X\rightarrow Y</math> every element of <math>X</math> is mapped t3 KB (463 words) - 21:50, 8 May 2018
- * The [[pre-image]] of every set open in {{M|Y}} under {{M|f}} is open in {{M|X}} ...,\mathcal{J})}} and {{M|(Y,\mathcal{K})}}, and a point {{M|x_0\in X}}, a [[function]] {{M|f:X\rightarrow Y}} is said to be ''continuous at {{M|x_0}}'' if<ref n6 KB (972 words) - 01:44, 14 October 2016
- * Need to add: [[A function is continuous if and only if the pre-image of every basis element is open]] - [[User:Alec|Alec]] ([[User talk:Alec|tal5 KB (802 words) - 18:35, 17 December 2016
- ...than this the references are valid, note that this is 'obvious' as if the image of {{M|\Vert\cdot\Vert}} could be in {{M|\mathbb{C} }} then the {{M|\Vert x Such a vector space equipped with such a function is called a [[Normed space|normed space]]<ref name="APIKM"/>6 KB (1,026 words) - 20:33, 9 April 2017
- ![[Image:Coordinate example.JPG|500px]] Yes that function <math>L(x,y)\mapsto(2x,2y)'</math> is actually <math>[Id]_S^{S'}</math> thi9 KB (1,525 words) - 16:30, 23 August 2015
- '''Note: ''' Sometimes called a ''measurable [[Function|fuction]]''<ref name="PAS">Probability and Stochastics - Erhan Cinlar</ref> ...ble 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|A map, {{M|f:(A,\mathcal{A})\rightarrow(F,\ma5 KB (792 words) - 02:31, 3 August 2015
- A parametrisation of a curve in {{M|\mathbb{R}^n}} is a function<ref> A parametrisation whos image is all (or a part of) a level curve is called a parameterisation (of part)1 KB (224 words) - 21:30, 28 March 2015
- Given a [[Continuous map|continuous]] and [[Bijection|bijective]] function between two [[Topological space|topological spaces]] {{M|f:X\rightarrow Y}} ::* (Using [[Image of a compact set is compact]])1 KB (219 words) - 12:36, 13 August 2015
- In fact given a function {{M|\times:S\times S\rightarrow S}} we even call the image of {{M|(x,y)}} under {{M|\times}} the ''product'' (or indeed the ''sum'' if455 B (77 words) - 07:44, 27 April 2015
- ...tyle=max-width:20em;}}Given a function, {{M|f:X\rightarrow Y}} and another function, {{M|w:X\rightarrow W}}<ref group="Note">I have chosen {{M|W}} to mean "wha ...P}(A)}} (a subset of {{M|A}}) we use {{M|f(D)}} to denote the {{link|image|function}} of {{M|D}} under {{M|f}}, namely: {{M|1=f(D):=\{f(d)\in B\ \vert\ d\in D\8 KB (1,644 words) - 20:49, 11 October 2016
- * [[Additive function]] '''DONE''' [[User:Alec|Alec]] ([[User talk:Alec|talk]]) 22:50, 19 March 2 * [[Integral of a simple function (measure theory)]] '''DONE''' [[User:Alec|Alec]] ([[User talk:Alec|talk]])5 KB (645 words) - 11:40, 21 August 2016
- * Image of a compact space is compact * A continuous and bijective function from a compact space to a Hausdorff space is a homeomorphism3 KB (616 words) - 08:37, 1 July 2015
- * Any [[Function|mapping]], {{M|1=f:X\rightarrow Y}}, is [[Continuous map|continuous]] ...thcal{K}[f^{-1}(A)\in\mathcal{J}]}} - for all open sets in {{M|Y}} the pre-image under {{M|f}} is open in {{M|X}}3 KB (534 words) - 13:07, 19 February 2016
- This page is intended to be a list of different conditions for a [[Function|function]] to be a [[Measurable map|measurable map]] ...ble 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|A map, {{M|f:(A,\mathcal{A})\rightarrow(F,\ma678 B (120 words) - 23:54, 2 August 2015
- * A chart is a function, {{M|\phi:U\rightarrow\mathbb{R}^n}} where {{M|U}} is some portion of our m * {{M|\text{Im}_\phi(U\cap V)}} is all the "image" of {{M|U\cap V}} under {{M|\phi}} and is the shaded bit on the bottom left10 KB (1,899 words) - 18:48, 23 September 2015
- Suppose we have a [[topological manifold]] {{M|M}} and a function {{M|f:M\rightarrow\mathbb{R} }} which is continuous. Notice that the notion (In this article {{M|A_\varphi}} will denote the image of {{M|A}} under the chart {{M|\varphi}})2 KB (414 words) - 12:26, 12 November 2015
- ...on that takes each element of the powerset of the domain of {{M|f}} to the image of that element under {{M|f}} (which is obviously in the powerset of the co ...Describe this, something like {{M|b}} is in the image of {{M|X}} under the image of {{M|f}} in the functor ''if and only if'' there is an {{M|a\in X}} such2 KB (317 words) - 17:51, 13 March 2016
- ...nt EVERY open set containing the image of that point must have an open pre-image. :Indeed, the pre-image need not be open (otherwise it is about continuity everywhere, not just at1 KB (238 words) - 20:15, 23 March 2016
- * [[Pre-image sigma-algebra]] - '''DONE''' - [[User:Alec|Alec]] ([[User talk:Alec|talk]]) * [[Pre-image sigma-algebra/Definition]] - '''DONE''' - [[User:Alec|Alec]] ([[User talk:A2 KB (212 words) - 00:09, 2 April 2016
- Suppose {{M|f:X\rightarrow Y}} is a [[function]] between two {{plural|topological space|s}}, {{Top.|X|J}} and {{Top.|Y|K}} {{Todo|1=I really need to create the pages that show the pre-image of functions preserves things like unions and intersections}}3 KB (467 words) - 16:58, 16 August 2016
