Editing Isomorphism (section)

==Relation to equality==
{{See also|Equality (mathematics)|coherent isomorphism}}

Although there are cases where isomorphic objects can be considered equal, one must distinguish  {{em|[[Equality (mathematics)|equality]]}} and {{em|isomorphism}}.<ref>{{Harvnb|Mazur|2007}}</ref> Equality is when two objects are the same, and therefore everything that is true about one object is true about the other. On the other hand, isomorphisms are related to some structure, and two isomorphic objects share only the properties that are related to this structure.

For example, the sets
<math display="block">A = \left\{ x \in \Z \mid x^2 < 2\right\} \quad \text{ and } \quad B = \{-1, 0, 1\}</math>
are {{em|equal}}; they are merely different representations—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 <math>\{A, B, C\}</math> and <math>\{1, 2, 3\}</math> are not {{em|equal}} since they do not have the same elements. They are isomorphic as sets, but there are many choices (in fact 6) of an isomorphism between them: one isomorphism is
:<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>
and no one isomorphism is intrinsically better than any other.<ref group="note"><math>A, B, C</math> have a conventional order, namely the alphabetical order, and similarly 1, 2, 3 have the usual order of the integers. Viewed as ordered sets, there is only one isomorphism between them, namely 
<math display="block">\text{A} \mapsto 1, \text{B} \mapsto 2, \text{C} \mapsto 3.</math></ref> On this view and in this sense, these two sets are not equal because one cannot consider them {{em|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.

Also, [[integer]]s and [[even number]]s are isomorphic as [[ordered set]]s and [[abelian group]]s (for addition), but cannot be considered equal sets, since one is a [[proper subset]] of the other.

On the other hand, when sets (or other [[mathematical object]]s) are defined only by their properties, without considering the nature of their elements, one often considers them to be equal. This is generally the case with solutions of [[universal properties]].

For example, the [[rational number]]s are formally defined as [[equivalence class]]es of pairs of integers, although nobody thinks of a rational number as a set (equivalence class). The universal property of the rational numbers is essentially that they form a [[field (mathematics)|field]] that contains the integers and does not contain any proper subfield. Given two fields with these properties, there is a unique field isomorphism between them. This allows identifying these two fields, since every property of one of them can be transferred to the other through the isomorphism. The [[real number]]s that can be expressed as a quotient of integers form the smallest subfield of the reals. There is thus a unique isomorphism from this subfield of the reals to the rational numbers defined by equivalence classes.