Proof of the inverse of an inverse

In summary, the conversation discusses how to verify two properties using the axioms in a group: that the inverse of the inverse of x is x, and that (xy)^-1 = y^-1x^-1. The conversation includes attempts at solving these problems and ultimately concludes that the proofs involve noticing that x is the inverse of x^-1.
  • #1
fk378
367
0

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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  • #2
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).
 
  • #3
So my proof should conclude with noticing that x is the inverse of x-inverse?
 
  • #4
fk378 said:
So my proof should conclude with noticing that x is the inverse of x-inverse?

Well, yes. It is, isn't it?
 
  • #5
For the second question - what happens if you multiply [tex] xy [/tex] with the object you need to show is its inverse?
 

1. What is the proof of the inverse of an inverse?

The proof of the inverse of an inverse is a mathematical concept that shows that the inverse of a number or function is itself. In other words, if you take the inverse of a number or function and then take the inverse of that result, you will end up with the original number or function.

2. Why is the proof of the inverse of an inverse important?

The proof of the inverse of an inverse is important because it helps verify the validity of mathematical operations involving inverses. It also helps in solving equations and understanding the properties of inverse functions.

3. How is the proof of the inverse of an inverse different from the proof of an inverse?

The proof of the inverse of an inverse is different from the proof of an inverse because it involves showing that the result of taking the inverse twice is the original number or function, whereas the proof of an inverse only shows that the inverse undoes the original operation.

4. Are there any exceptions to the proof of the inverse of an inverse?

No, the proof of the inverse of an inverse holds true for all numbers and functions except for when the original number or function is undefined or has no inverse.

5. Can the proof of the inverse of an inverse be applied to all types of inverses?

Yes, the proof of the inverse of an inverse can be applied to all types of inverses, including multiplicative and additive inverses. It is a general mathematical concept that applies to any operation that has an inverse.

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