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matematikuvol
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Homomorphism is defined by ##f(x*y)=f(x)\cdot f(y)##. One interesting example of this is logarithm function ##log(xy)=\log x+\log y##. Can you explain me why this is also isomorphism?
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Can you explain me why this is also isomorphism?
- Thread starter matematikuvol
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- Explain Isomorphism
In summary, homomorphism is a function defined by ##f(x*y)=f(x)\cdot f(y)##. An interesting example is the logarithm function, where ##\log(xy)=\log x+\log y##. This is also an isomorphism, which is a bijective homomorphism. However, the function ##f(x)=e^x## is not surjective unless it is restricted to only positive numbers.
- #2
tiny-tim
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hi matematikuvol! 
an isomorphism is a homomorphism that is bijective (one-to-one and onto)
(see http://en.wikipedia.org/wiki/Isomorphism)
an isomorphism is a homomorphism that is bijective (one-to-one and onto)
(see http://en.wikipedia.org/wiki/Isomorphism)
- #3
matematikuvol
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I know that. But I asked only for logarithm because
[tex]\log (ab)=\log (a)+\log (b)[/tex]
[tex]\log ((-a)(-b))=\log (a)+\log (b)[/tex]
Why function ##f(x)=e^x## isn't surjective?
- #4
tiny-tim
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it's only an isomorphism if you restrict it to the positive numbers
(see also http://en.wikipedia.org/wiki/Isomorphism#Practical_examples)
(see also http://en.wikipedia.org/wiki/Isomorphism#Practical_examples)
- #5
matematikuvol
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Ok. Tnx for the answer.
1. Can you define what an isomorphism is?
An isomorphism is a mathematical concept that describes a one-to-one correspondence between two objects or structures. In other words, it is a mapping or transformation that preserves certain properties between two systems.
2. How is an isomorphism different from an equivalence relation?
While both an isomorphism and an equivalence relation establish a relationship between two objects, an isomorphism specifically refers to a structure-preserving relationship, while an equivalence relation refers to a relationship based on similarity or equality.
3. What are some examples of isomorphism in real life?
Isomorphism can be seen in various fields, such as chemistry, biology, and computer science. For example, the human hand and a bat's wing have a similar structure, which can be described as an isomorphism. In chemistry, the isomorphism between different crystal structures of a substance can determine its properties.
4. How is isomorphism useful in the scientific community?
Isomorphism allows scientists to better understand and analyze complex systems by identifying and comparing their underlying structures. It also helps in making predictions and generalizations about similar systems.
5. Can two objects have more than one isomorphism between them?
Yes, it is possible for two objects to have multiple isomorphisms between them. In some cases, these isomorphisms may be equivalent, while in others, they may differ in their level of preservation of structure. For example, the isomorphism between two graphs can be based on their number of edges or their degree of connectivity.
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