Isomorphism between II18 / <3> and II3

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In summary, the isomorphism problem is a mathematical problem that involves determining whether two structures or objects are essentially the same, despite having different descriptions or representations. It has applications in various fields, including mathematics, computer science, and chemistry. Examples of the isomorphism problem include comparing graphs, groups, and chemical compounds. It is important because it allows us to understand the underlying structure of objects and has practical applications. Isomorphism and homomorphism are related concepts, with isomorphism referring to the equality of two structures and homomorphism referring to a mapping between two structures that preserves certain properties. While there is no general solution to the isomorphism problem, there are specific algorithms and techniques that can be used to compare specific structures
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aliciaislol
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Homework Statement


Show that II18 / <3> is isomorphic to II3.


Homework Equations


II18 = {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18}
<3> = {0,3,6,9,12,15}
II3 = {1,2,3}
II18 / <3> = {3,6,9,12,15,18}

The Attempt at a Solution

 
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aliciaislol said:

Homework Statement


Show that II18 / <3> is isomorphic to II3.


Homework Equations


II18 = {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18}
<3> = {0,3,6,9,12,15}
II3 = {1,2,3}
II18 / <3> = {3,6,9,12,15,18}

The Attempt at a Solution

Looks to me like your proof says they are NOT isomorphic! If they don't have the same number of members, they certainly aren't isomorphic.
 

What is the isomorphism problem?

The isomorphism problem is a mathematical problem that asks whether two structures or objects are essentially the same, despite having different descriptions or representations. It is a fundamental problem in many areas of mathematics, computer science, and chemistry.

What are some examples of the isomorphism problem?

Examples of the isomorphism problem include determining whether two graphs are isomorphic, whether two groups are isomorphic, and whether two chemical compounds have the same structure despite having different molecular formulas.

Why is the isomorphism problem important?

The isomorphism problem is important because it allows us to identify and understand the underlying structure of objects or systems. It has applications in areas such as cryptography, pattern recognition, and the classification of chemical compounds.

What is the difference between isomorphism and homomorphism?

Isomorphism and homomorphism are related concepts, but they have different definitions. Isomorphism refers to the equality of two structures, while homomorphism refers to a mapping between two structures that preserves certain properties, such as operations or relationships.

Is there a solution to the isomorphism problem?

The isomorphism problem is a difficult problem with no general solution. However, there are algorithms and techniques that can be used to determine whether two specific structures or objects are isomorphic. These solutions are often specific to the type of structures being compared.

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