G-equidecomposable and equivalence relation

In summary, the conversation discusses the relation A~B being an equivalence relation on subsets of X if and only if A and B are G-equidecomposable. This requires checking three properties for equidecomposability, which are generally trivial for groups due to their inverse, identity, and associative properties. The Banach-Schroder theorem may also be applicable in this situation.
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JSG31883
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G acts via isometries on a set X, and A,B are subsets of X. Prove that the relation A~B is an equivalence relation on subsets of X iff A and B are G-equidecomposable.


I think this has to do with the Banach-Schroder theorem, but am not sure. I know it is a definition in group theory, but am not sure how to prove it since it seem pretty self explanitory to me.
 
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  • #2
Can someone help?
 
  • #3
define G-equidecomposable, please,

the result should be easy isnce it just checking 3 things for the property of equidecomposability, whatever that may be, which are generally trivial for groups since groups have inverses, and identity and composition is associative. eg if X~Y and g in G effects this relation, then g^{-1} will (probably) effect the relation Y~X, if X~Y and Y~Z and the relation is because of elements f and g resp. then gf wil mean that X~Y, and X~X because e(X)=X.

note i haven't a clue what equidecompsoability is, but this will be the proof, I'm almost sure of it.
 

1. What is G-equidecomposability?

G-equidecomposability is a mathematical concept that refers to the ability to partition a set into smaller subsets in such a way that each subset can be transformed into the other using a specific group of operations. This concept is often used in algebraic topology and geometry.

2. How is G-equidecomposability related to equivalence relations?

G-equidecomposability and equivalence relations are closely related because both concepts involve partitioning a set into subsets. However, G-equidecomposability specifically looks at partitioning a set using a specific group of operations, while equivalence relations focus on partitioning a set based on a specific rule or property.

3. Can you give an example of G-equidecomposability?

One example of G-equidecomposability is the decomposition of a square into two triangles. This can be achieved by rotating or reflecting one of the triangles. In this case, the group of operations is the set of rotations and reflections.

4. What are the applications of G-equidecomposability?

G-equidecomposability has applications in various fields, including algebraic topology, geometry, and physics. It is used to study the structure of spaces and objects and to understand how they can be transformed through different operations.

5. How is G-equidecomposability different from G-equivalence?

G-equidecomposability and G-equivalence are related but different concepts. G-equidecomposability focuses on the ability to partition a set into smaller subsets using a specific group of operations, while G-equivalence refers to the similarity or equivalence between two objects under a specific group of operations. In other words, G-equidecomposability deals with the partitioning of a set, while G-equivalence deals with the relationship between two objects.

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