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| {{About|mathematics}}
| | Hello brother. Let me introduce myself. I am Cassandra though I don't really like being called like why. Kansas is where his residence is. What me and our grandkids love has been doing 3d graphics and I'm trying truly a field. Since I was 18 I've been working the office supervisor but the promotion never comes.<br><br>My blog post; [http://mohamedia.net/ Decorate your home] |
| {{multiple image
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| | footer = The group of fifth roots of unity under multiplication is isomorphic to the group of rotations of the pentagon under composition.
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| | width = 200
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| | image1 = One5Root.svg
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| | alt1 = Fifth roots of unity
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| | image2 = Regular polygon 5 annotated.svg
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| | alt2 = Rotations of a pentagon
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| }}
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| In [[mathematics]], an '''isomorphism''', from the [[Greek language|Greek]]: [[wikt:ἴσος|ἴσος]] ''isos'' "equal", and [[wikt:μορφή|μορφή]] ''morphe'' "shape", is a [[homomorphism]] (or more generally a [[morphism]]) that admits an inverse.<ref group=note>For clarity, by ''inverse'' is meant ''inverse homomorphism'' or ''inverse morphism'' respectively, not ''inverse function''.</ref> Two [[mathematical object]]s are '''isomorphic''' if an isomorphism exists between them. An ''[[automorphism]]'' is an isomorphism whose source and target coincide. The interest of isomorphisms lies in the fact that two isomorphic objects cannot be distinguished by using only the properties used to define morphisms; thus isomorphic objects may be considered the same as long as one considers only these properties and their consequences.
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| For most [[algebraic structure]]s, including [[group (mathematics)|group]]s and [[ring (mathematics)|ring]]s, a homomorphism is an isomorphism if and only if it is [[bijective]].
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| In [[topology]], where the morphisms are [[continuous function]]s, isomorphisms are also called ''[[homeomorphism]]s'' or ''bicontinuous functions''. In [[mathematical analysis]], where the morphisms are [[differentiable function]]s, isomorphisms are also called ''[[diffeomorphism]]s''.
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| Isomorphisms are formalized using [[category theory]]. A morphism {{nowrap|''f'' : ''X'' → ''Y''}} in a category is an isomorphism if it admits a two-sided inverse, meaning that there is another morphism {{nowrap|''g'' : ''Y'' → ''X''}} in that category such that {{nowrap|''gf'' {{=}} 1<sub>''X''</sub>}} and {{nowrap|''fg'' {{=}} 1<sub>''Y''</sub>}}, where 1<sub>''X''</sub> and 1<sub>''Y''</sub> are the identity morphisms of ''X'' and ''Y'', respectively.<ref>{{cite book|author=Awodey, Steve|chapter=Isomorphisms|title=Category theory|publisher=Oxford University Press|year=2006|isbn=9780198568612|page=11|url=http://books.google.com/books?id=IK_sIDI2TCwC&pg=PA11}}</ref>
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| ==Examples==<!-- This section is linked from [[List of small groups]] -->
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| ===Logarithm and exponential===
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| Let <math>\mathbb{R}^\times_{>0}</math> be the multiplicative group of positive real numbers, and let <math>\mathbb{R}</math> be the additive group of real numbers. | |
| The [[logarithm function]] <math>\log \colon \mathbb{R}^\times_{>0} \to \mathbb{R}</math> satisfies <math>\log(xy) = \log x + \log y</math> for all <math>x,y \in \mathbb{R}^\times_{>0}</math>, so it is a [[group homomorphism]]. The [[exponential function]] <math>\exp \colon \mathbb{R} \to \mathbb{R}^\times_{>0}</math> satisfies <math>\exp(x+y) = (\exp x)(\exp y)</math> for all <math>x,y \in \mathbb{R}</math>, so it too is a homomorphism. The identities <math>\log \exp x = x</math> and <math>\exp \log y = y</math> show that <math>\log</math> and <math>\exp</math> are inverses of each other. Since <math>\log</math> is a homomorphism that has an inverse that is also a homomorphism, <math>\log</math> is an isomorphism of groups.
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| Because <math>\log</math> is an isomorphism, it translates multiplication of positive real numbers into addition of real numbers. This is what makes it possible to multiply real numbers using a [[ruler]] and a [[table of logarithms]], or using a [[slide rule]] with a logarithmic scale.
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| ===Integers modulo 6===
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| Consider the group <math>(\mathbb{Z}_6, +)</math>, the integers from 0 to 5 with addition [[modular arithmetic|modulo]] 6. Also consider the group <math>(\mathbb{Z}_2 \times \mathbb{Z}_3, +)</math>, the ordered pairs where the ''x'' coordinates can be 0 or 1, and the y coordinates can be 0, 1, or 2, where addition in the ''x''-coordinate is modulo 2 and addition in the ''y''-coordinate is modulo 3.
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| These structures are isomorphic under addition, if you identify them using the following scheme:
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| :(0,0) → 0
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| :(1,1) → 1
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| :(0,2) → 2
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| :(1,0) → 3
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| :(0,1) → 4
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| :(1,2) → 5
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| or in general (''a'',''b'') → (3''a'' + 4''b'') mod 6.
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| For example note that (1,1) + (1,0) = (0,1), which translates in the other system as 1 + 3 = 4.
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| Even though these two groups "look" different in that the sets contain different elements, they are indeed '''isomorphic''': their structures are exactly the same. More generally, the [[direct product of groups|direct product]] of two [[cyclic group]]s <math>\mathbb{Z}_m</math> and <math>\mathbb{Z}_n</math> is isomorphic to <math>(\mathbb{Z}_{mn}, +)</math> if and only if ''m'' and ''n'' are [[coprime]].
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| ===A relation-preserving isomorphism===
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| If one object consists of a set ''X'' with a [[binary relation]] R and the other object consists of a set ''Y'' with a binary relation S then an isomorphism from ''X'' to ''Y'' is a bijective function {{nowrap|1=ƒ: ''X'' → ''Y''}} such that:<ref>{{Cite book|author=Vinberg, Ėrnest Borisovich|title=A Course in Algebra|publisher=American Mathematical Society|year=2003|isbn=9780821834138|page=3|url=http://books.google.com/books?id=kd24d3mwaecC&pg=PA3}}</ref>
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| :<math> \operatorname{S}(f(u),f(v)) \iff \operatorname{R}(u,v) </math>
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| S is [[reflexive relation|reflexive]], [[irreflexive relation|irreflexive]], [[symmetric relation|symmetric]], [[antisymmetric relation|antisymmetric]], [[asymmetric relation|asymmetric]], [[transitive relation|transitive]], [[total relation|total]], [[Binary relation#Relations over a set|trichotomous]], a [[partial order]], [[total order]], [[strict weak order]], [[Strict weak order#Total preorders|total preorder]] (weak order), an [[equivalence relation]], or a relation with any other special properties, if and only if R is.
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| For example, R is an [[Order theory|ordering]] ≤ and S an ordering <math>\scriptstyle \sqsubseteq</math>, then an isomorphism from ''X'' to ''Y'' is a bijective function {{nowrap|1=ƒ: ''X'' → ''Y''}} such that
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| :<math>f(u) \sqsubseteq f(v) \iff u \le v . </math>
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| Such an isomorphism is called an ''[[order isomorphism]]'' or (less commonly) an ''isotone isomorphism''.
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| If {{nowrap|1=''X'' = ''Y''}}, then this is a relation-preserving [[automorphism]].
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| ==Isomorphism vs. bijective morphism==
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| In a [[concrete category]] (that is, roughly speaking, a category whose objects are sets and morphisms are mappings between sets), such as the category of topological spaces or categories of algebraic objects like groups, rings, and modules, an isomorphism must be bijective on the underlying sets. In algebraic categories (specifically, categories of [[variety (universal algebra)|varieties in the sense of universal algebra]]), an isomorphism is the same as a homomorphism which is bijective on underlying sets. However, there are concrete categories in which bijective morphisms are not necessarily isomorphisms (such as the category of topological spaces), and there are categories in which each object admits an underlying set but in which isomorphisms need not be bijective (such as the homotopy category of CW-complexes).
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| ==Applications==
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| In [[abstract algebra]], two basic isomorphisms are defined:
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| * [[Group isomorphism]], an isomorphism between [[group (mathematics)|groups]]
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| * [[Ring isomorphism]], an isomorphism between [[ring (mathematics)|rings]]. (Note that isomorphisms between [[field (mathematics)|fields]] are actually ring isomorphisms)
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| Just as the [[automorphism]]s of an [[algebraic structure]] form a [[group (mathematics)|group]], the isomorphisms between two algebras sharing a common structure form a [[heap (mathematics)|heap]]. Letting a particular isomorphism identify the two structures turns this heap into a group.
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| In [[mathematical analysis]], the [[Laplace transform]] is an isomorphism mapping hard [[differential equations]] into easier [[algebra]]ic equations.
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| In [[category theory]], Iet the [[category (mathematics)|category]] ''C'' consist of two [[class (set theory)|classes]], one of ''objects'' and the other of [[morphisms]]. Then a general definition of isomorphism that covers the previous and many other cases is: an isomorphism is a morphism {{nowrap|1=ƒ: ''a'' → ''b''}} that has an inverse, i.e. there exists a morphism {{nowrap|1=''g'': ''b'' → ''a''}} with {{nowrap|1=''ƒg'' = 1<sub>''b''</sub>}} and {{nowrap|1=''gƒ'' = 1<sub>''a''</sub>}}. For example, a bijective [[linear map]] is an isomorphism between [[vector space]]s, and a bijective [[continuous function]] whose inverse is also continuous is an isomorphism between [[topological space]]s, called a [[homeomorphism]].
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| In [[graph theory]], an isomorphism between two graphs ''G'' and ''H'' is a [[bijective]] map ''f'' from the vertices of ''G'' to the vertices of ''H'' that preserves the "edge structure" in the sense that there is an edge from [[vertex (graph theory)|vertex]] ''u'' to vertex ''v'' in ''G'' if and only if there is an edge from ƒ(''u'') to ƒ(''v'') in ''H''. See [[graph isomorphism]].
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| In mathematical analysis, an isomorphism between two [[Hilbert spaces]] is a bijection preserving addition, scalar multiplication, and inner product.
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| In early theories of [[logical atomism]], the formal relationship between facts and true propositions was theorized by [[Bertrand Russell]] and [[Ludwig Wittgenstein]] to be isomorphic. An example of this line of thinking can be found in Russell's [[Introduction to Mathematical Philosophy]].
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| In [[cybernetics]], the [[Good Regulator]] or Conant-Ashby theorem is stated "Every Good Regulator of a system must be a model of that system". Whether regulated or self-regulating an isomorphism is required between regulator part and the processing part of the system.
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| ==Relation with equality==
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| {{See also|Equality (mathematics)}}
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| In certain areas of mathematics, notably [[category theory]], it is valuable to distinguish between ''[[Equality (mathematics)|equality]]'' on the one hand and ''isomorphism'' on the other.<ref>{{Harvnb|Mazur|2007}}</ref> Equality is when two objects are exactly the same, and everything that's true about one object is true about the other, while an isomorphism implies everything that's true about a designated part of one object's structure is true about the other's. For example, the sets
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| :<math>A = \{ x \in \mathbb{Z} \mid x^2 < 2\}</math> and <math>B = \{-1, 0, 1\} \,</math>
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| are ''equal''; they are merely different presentations—the first an [[intensional definition|intensional]] one (in [[set builder notation]]), and the second [[extensional definition|extensional]] (by explicit enumeration)—of the same subset of the integers. By contrast, the sets {''A'',''B'',''C''} and {1,2,3} are not ''equal''—the first has elements that are letters, while the second has elements that are numbers. These are isomorphic as sets, since finite sets are determined up to isomorphism by their [[cardinality]] (number of elements) and these both have three elements, but there are many choices of isomorphism—one isomorphism is
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| :<math>\text{A} \mapsto 1, \text{B} \mapsto 2, \text{C} \mapsto 3,</math> while another is <math>\text{A} \mapsto 3, \text{B} \mapsto 2, \text{C} \mapsto 1,</math>
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| and no one isomorphism is intrinsically better than any other.<ref group="note">The careful reader may note that ''A'', ''B'', ''C'' have a conventional order, namely alphabetical order, and similarly 1, 2, 3 have the order from the integers, and thus one particular isomorphism is "natural", namely
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| :<math>\text{A} \mapsto 1, \text{B} \mapsto 2, \text{C} \mapsto 3</math>.
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| More formally, as ''sets'' these are isomorphic, but not naturally isomorphic (there are multiple choices of isomorphism), while as ''ordered sets'' they are naturally isomorphic (there is a unique isomorphism, given above), since [[finite total order]]s are uniquely determined up to unique isomorphism by [[cardinality]].
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| This intuition can be formalized by saying that any two finite [[totally ordered set]]s of the same cardinality have a natural isomorphism, the one that sends the [[least element]] of the first to the least element of the second, the least element of what remains in the first to the least element of what remains in the second, and so forth, but in general, pairs of sets of a given finite cardinality are not naturally isomorphic because there is more than one choice of map—except if the cardinality is 0 or 1, where there is a unique choice.</ref><ref group="note">In fact, there are precisely <math>3! = 6</math> different isomorphisms between two sets with three elements. This is equal to the number of [[automorphism]]s of a given three-element set (which in turn is equal to the order of the [[symmetric group]] on three letters), and more generally one has that the set of isomorphisms between two objects, denoted <math>\operatorname{Iso}(A,B),</math> is a [[torsor]] for the automorphism group of ''A,'' <math>\operatorname{Aut}(A)</math> and also a torsor for the automorphism group of ''B.'' In fact, automorphisms of an object are a key reason to be concerned with the distinction between isomorphism and equality, as demonstrated in the effect of change of basis on the identification of a vector space with its dual or with its double dual, as elaborated in the sequel.</ref> On this view and in this sense, these two sets are not equal because one cannot consider them ''identical'': one can choose an isomorphism between them, but that is a weaker claim than identity—and valid only in the context of the chosen isomorphism.
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| Sometimes the isomorphisms can seem obvious and compelling, but are still not equalities. As a simple example, the [[genealogy|genealogical]] relationships among [[Joseph Kennedy|Joe]], [[John F. Kennedy|John]], and [[Robert F. Kennedy|Bobby]] Kennedy are, in a real sense, the same as those among the [[American football]] [[quarterbacks]] in the Manning family: [[Archie Manning|Archie]], [[Peyton Manning|Peyton]], and [[Eli Manning|Eli]]. The father-son pairings and the elder-brother-younger-brother pairings correspond perfectly. That similarity between the two family structures illustrates the origin of the word ''isomorphism'' (Greek ''iso''-, "same," and -''morph'', "form" or "shape"). But because the Kennedys are not the same people as the Mannings, the two genealogical structures are merely isomorphic and not equal.
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| Another example is more formal and more directly illustrates the motivation for distinguishing equality from isomorphism: the distinction between a [[finite-dimensional vector space]] ''V'' and its [[dual space]] {{nowrap|1=''V''* = { φ: V → '''K''' }}} of linear maps from ''V'' to its field of scalars '''K'''.
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| These spaces have the same dimension, and thus are isomorphic as abstract vector spaces (since algebraically, vector spaces are classified by dimension, just as sets are classified by cardinality), but there is no "natural" choice of isomorphism <math>\scriptstyle V \, \overset{\sim}{\to} \, V^*</math>.
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| If one chooses a basis for ''V'', then this yields an isomorphism: For all {{nowrap|1=''u''. ''v'' ∈ ''V''}},
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| :<math>v \ \overset{\sim}{\mapsto} \ \phi_v \in V^* \quad \text{such that} \quad \phi_v(u) = v^\mathrm{T} u</math>.
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| This corresponds to transforming a [[column vector]] (element of ''V'') to a [[row vector]] (element of ''V''*) by [[transpose]], but a different choice of basis gives a different isomorphism: the isomorphism "depends on the choice of basis".
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| More subtly, there ''is'' a map from a vector space ''V'' to its [[double dual]] {{nowrap|1= ''V''** = { ''x'': ''V''* → '''K''' }}} that does not depend on the choice of basis: For all {{nowrap|1=''v'' ∈ ''V'' and φ ∈ ''V''*,}}
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| :<math>v \ \overset{\sim}{\mapsto} \ x_v \in V^{**} \quad \text{such that} \quad x_v(\phi) = \phi(v)</math>.
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| This leads to a third notion, that of a [[natural isomorphism]]: while ''V'' and ''V''** are different sets, there is a "natural" choice of isomorphism between them.
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| This intuitive notion of "an isomorphism that does not depend on an arbitrary choice" is formalized in the notion of a [[natural transformation]]; briefly, that one may ''consistently'' identify, or more generally map from, a vector space to its double dual, <math>\scriptstyle V \, \overset{\sim}{\to} \, V^{**}</math>, for ''any'' vector space in a consistent way.
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| Formalizing this intuition is a motivation for the development of category theory.
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| If one wishes to draw a distinction between an arbitrary isomorphism (one that depends on a choice) and a natural isomorphism (one that can be done consistently), one may write <big>≈</big> for an [[unnatural isomorphism]] and ≅ for a natural isomorphism, as in {{nowrap|1=''V'' <big>≈</big> ''V''*}} and {{nowrap|1=''V'' ≅ ''V''**.}}
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| This convention is not universally followed, and authors who wish to distinguish between unnatural isomorphisms and natural isomorphisms will generally explicitly state the distinction.
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| Generally, saying that two objects are ''equal'' is reserved for when there is a notion of a larger (ambient) space that these objects live in. Most often, one speaks of equality of two subsets of a given set (as in the integer set example above), but not of two objects abstractly presented. For example, the 2-dimensional unit sphere in 3-dimensional space
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| :<math>S^2 := \{ (x,y,z) \in \mathbb{R}^3 \mid x^2 + y^2 + z^2 = 1\}</math> and the [[Riemann sphere]] <math>\widehat{\mathbb{C}}</math>
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| which can be presented as the [[one-point compactification]] of the complex plane {{nowrap|1='''C''' ∪ {∞}}} ''or'' as the complex [[projective line]] (a [[quotient space]])
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| :<math>\mathbf{P}_{\mathbb{C}}^1 := (\mathbb{C}^2\setminus \{(0,0)\}) / (\mathbb{C}^*)</math>
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| are three different descriptions for a mathematical object, all of which are isomorphic, but not ''equal'' because they are not all subsets of a single space: the first is a subset of '''R'''<sup>3</sup>, the second is {{nowrap|1='''C''' ≅ '''R'''}}<sup>2</sup><ref group="note">Being precise, the identification of the complex numbers with the real plane,
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| :<math>\mathbf{C} \cong \mathbf{R}\cdot 1 \oplus \mathbf{R} \cdot i = \mathbf{R}^2</math>
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| depends on a choice of <math>i;</math> one can just as easily choose <math>(-i),</math>, which yields a different identification—formally, [[complex conjugation]] is an automorphism—but in practice one often assumes that one has made such an identification.</ref> plus an additional point, and the third is a [[subquotient]] of '''C'''<sup>2</sup>
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| In the context of category theory, objects are usually at most isomorphic—indeed, a motivation for the development of category theory was showing that different constructions in [[homology theory]] yielded equivalent (isomorphic) groups. Given maps between two objects ''X'' and ''Y'', however, one asks if they are equal or not (they are both elements of the set Hom(''X'', ''Y''), hence equality is the proper relationship), particularly in [[commutative diagram]]s.
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| ==See also==
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| {{Portal|Mathematics}}
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| *[[Bisimulation]]
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| *[[Epimorphism]]
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| *[[Heap (mathematics)]]
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| *[[Isometry]]
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| *[[Isomorphism class]]
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| *[[Isomorphism theorem]]
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| *[[Monomorphism]]
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| ==Notes==
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| {{Reflist|group=note}}
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| ==References==
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| {{refimprove|date=September 2010}}
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| {{Reflist}}
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| ==Further reading==
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| * {{Citation | first = Barry | last = Mazur | authorlink = Barry Mazur | title = When is one thing equal to some other thing? | date = 12 June 2007 | url = http://www.math.harvard.edu/~mazur/preprints/when_is_one.pdf | ref = harv}}
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| ==External links==
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| {{Wiktionary|isomorphism}}
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| *{{springer|title=Isomorphism|id=p/i052840}}
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| *{{planetmath reference|id=1936|title=Isomorphism}}
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| *{{MathWorld | urlname=Isomorphism | title = Isomorphism}}
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| [[Category:Morphisms]]
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