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Isomorphism - Practical example | | Isomorphism - Practical example: 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 | | | 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 | | |
| | | Isomorphism: Encyclopedia II - Isomorphism - Practical example
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 obeys the following identity:
But the real numbers under addition also form a group. So the logarithm function is in fact a group isomorphism from the group to the group .
Other related archivesAbstract algebra, Algebra, Analysis, Big Ben, Category theory, Douglas Hofstadter, Eilhard Mitscherlich, Greek, Group isomorphism, Gödel, Escher, Bach, Legendre transform, algebra, algebraic, automorphism, bijection, bijective, binary operations, category theory, codomain, differential equations, domain, epimorphism, fields, graph theory, group, groups, homomorphism, homomorphisms, iff, inverse, isomorphism (sociology), isomorphism class, linear algebra, linear map, logarithm, mapping, mathematics, monomorphism, morphism, one-to-one, onto, order isomorphism, range, real numbers, sets, universal algebra, vector spaces, vertex
 Adapted from the Wikipedia article "Practical example", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki |
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