Proof of the inverse of an inverse

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    Inverse Proof
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Homework Help Overview

The discussion revolves around verifying properties of inverses in group theory, specifically focusing on two statements: the inverse of the inverse of an element and the inverse of a product of two elements.

Discussion Character

  • Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants explore the axioms of group theory to verify the properties of inverses, with some questioning how to proceed from established axioms. Others suggest considering the implications of the definitions of inverses.

Discussion Status

Some participants have offered insights into the relationships between elements and their inverses, while others are still grappling with how to formulate their proofs. There is an ongoing exploration of the implications of the axioms without a clear consensus on the final proofs.

Contextual Notes

Participants express uncertainty about how to begin their proofs and the implications of the definitions of inverses, indicating a need for further clarification on the axioms involved.

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Homework Statement


In any group, verify directly from the axioms that
(a) inverse of the inverse of x= x
(b) (xy)^inverse = (inverse y)(inverse x) for all x,y in G. (note the reversal here)


The Attempt at a Solution


(a) I tried to use the axiom that xe=x=ex but I don't know where to go from there.
(b) I don't know how to start it.
 
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If b is the inverse of a then ab=ba=e. If a is the inverse of b then ba=ab=e. They are the SAME THING. Think of what that means if a=x and b=x^(-1).
 
So my proof should conclude with noticing that x is the inverse of x-inverse?
 
fk378 said:
So my proof should conclude with noticing that x is the inverse of x-inverse?

Well, yes. It is, isn't it?
 
For the second question - what happens if you multiply [tex]xy[/tex] with the object you need to show is its inverse?
 

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