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Isomorphism - Definition | A Wisdom Archive on Isomorphism - Definition | | Isomorphism - Definition A selection of articles related to Isomorphism - Definition | |
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More material related to Isomorphism can be found here:
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Isomorphism, Isomorphism - A relation-preserving isomorphism, Isomorphism - An operation-preserving isomorphism, Isomorphism - Applications, Isomorphism - Definition, Isomorphism - Physical analogies, Isomorphism - Practical example, Isomorphism - Purpose, Isomorphism - Two abstract examples, automorphism, homomorphism, epimorphism, isomorphism class, monomorphism, morphism
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ARTICLES RELATED TO Isomorphism - Definition | | | |  Isomorphism - Definition: Encyclopedia II - Isomorphism - DefinitionDouglas Hofstadter provides an informal definition:
The word "isomorphism" applies when two complex structures can be mapped onto each other, in such a way that to each part of one structure there is a corresponding part in the other structure, where "corresponding" means that the two parts play similar roles in their respective structures. (Gödel, Escher, Bach, p. 49)
Formally, an isomorphism is a bijective map f such that both f and its inverse f −1 are homomorphisms, ...
See also:Isomorphism, Isomorphism - Definition, Isomorphism - Purpose, Isomorphism - Physical analogies, Isomorphism - Practical example, Isomorphism - Two abstract examples, Isomorphism - A relation-preserving isomorphism, Isomorphism - An operation-preserving isomorphism, Isomorphism - Applications Read more here: » Isomorphism: Encyclopedia II - Isomorphism - Definition |
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| | | | Isomorphism - Definition: Encyclopedia II - Isomorphism - Applications Group isomorphism is where the objects in question are groups. Similarly, if the objects are fields, it is called a field isomorphism.
In Analysis, the Legendre transform maps hard differential equations into easier algebraic equations.
In universal algebra, one can provide a general definition of isomorphism that covers these and many other cases. For a more general definition, see category theory.
In graph theory, an isomorphism between two graphs G and H is a bijective map f from the ...
See also:Isomorphism, Isomorphism - Definition, Isomorphism - Purpose, Isomorphism - Physical analogies, Isomorphism - Practical example, Isomorphism - Two abstract examples, Isomorphism - A relation-preserving isomorphism, Isomorphism - An operation-preserving isomorphism, Isomorphism - Applications Read more here: » Isomorphism: Encyclopedia II - Isomorphism - Applications |
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| | | | Isomorphism - Definition: Encyclopedia II - Isomorphism - Practical example The following is an example of an isomorphism from ordinary algebra.
Consider the logarithm function: For any fixed base b, the logarithm function logb maps from the positive real numbers onto the real numbers ; formally:
This mapping is one-to-one and onto, that is, it is a bijection from the domain to the codomain of the logarithm function.
In addition to being an isomorphism of sets, the logarithm function also preserves certain operations. Specifically, consider the group of positive real numbers under ordinary multiplication. The logarithm function o ...
See also:Isomorphism, Isomorphism - Definition, Isomorphism - Purpose, Isomorphism - Physical analogies, Isomorphism - Practical example, Isomorphism - Two abstract examples, Isomorphism - A relation-preserving isomorphism, Isomorphism - An operation-preserving isomorphism, Isomorphism - Applications Read more here: » Isomorphism: Encyclopedia II - Isomorphism - Practical example |
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