Inverse composite proof (wording of the proof)

In summary, given invertible mappings f:A->B and g:B->C, we want to prove that (g o f)^-1 = f^-1 o g^-1 for all c in C. This can be done by showing that, for any arbitrary c in C, (g o f)^-1(c) = f^-1(g^-1(c)). Using the definitions of f^-1 and g^-1, we can see that f^-1(g^-1(c)) = a, and since c is arbitrary, it suffices to show that (g o f)^-1(c) = a. This can be done by showing that, for any a in A, there exists a unique b in B and
  • #1
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Homework Statement


Let f:A->B and g:B->C be invertible mappings. Show (g o f)^-1 = f^-1 o g^-1.


Homework Equations


A mapping is invertible iff it is bijective


The Attempt at a Solution


I understand why these are equivalent statements; however, I can't figure out the wording of the proof.

The best I can think of is:

Suppose f and g are invertible mappings defined f:A->B and g:B->C. Let a, b, and c be elements of sets A, B, and C respectively such that f(a)=b and g(b)=c. Since f and g are bijective, f^-1(b)=a and g^-1(c)=b. So (g o f)^-1(c) = a = f^-1(g^-1(c)).
 
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  • #2
(g o f)^-1(c) = a

You haven't actually proven this to be the case.We want to show for all [itex]c\in C[/itex] that [itex](g\circ f)^{-1}(c) = f^{-1}(g^{-1}(c))[/itex]. Let [itex]b\in B[/itex] be the unique element of B such that [itex] g^{-1}(c)=b[/itex] and [itex]a\in A[/itex] the unique element of A such that [itex]f^{-1}(b) = a[/itex]. These both exist because f and g are invertible.

Clearly [itex]f^{-1}(g^{-1}(c)) = a[/itex] just by how a and b were defined. Since c is arbitrary, it suffices to prove that [itex](g\circ f)^{-1}(c) = a[/itex] as well. How can you do that?
 
  • #3
"Since c is arbitrary, it suffices to prove that (g∘f)−1(c)=a as well. How can you do that?"

I don't quiet understand your question, or the statement before it.
 
  • #4
what other statement involving a and c can you use, here?

where did a and c come from?

can we get c from a, somehow using f and g? how?

our assumptions have been as follows:

g-1:c-->b

f-1:b-->a

what is f(a)?
what is g(b)?
 
  • #5
psycho2499 said:
"Since c is arbitrary, it suffices to prove that (g∘f)−1(c)=a as well. How can you do that?"

I don't quiet understand your question, or the statement before it.


To prove that two functions a(x) and b(x) are equal, you can prove that given any possible input x0, a(x0)=b(x0) (this shouldn't be particularly surprising) So to prove that [itex] (g\circ f)^{-1} = f^{-1}\circ g^{-1}[/itex], you can prove that given any [itex]c\in C[/itex], [itex] (g\circ f)^{-1}(c) = f^{-1}(g^{-1}(c))[/itex]. We know that the right hand side of this last equation is a by how we defined b and a in my post... how can you prove that the left hand side is equal to a as well?
 
  • #6
This is the part that confuses me. Should I be saying something to the effect:
For any a[itex]\in[/itex]A there exists a unique b[itex]\in[/itex]B and a unique c[itex]\in[/itex]C such that f(a)=b and g(b)=c. It follows that g(f(a))=c. Thus (g o f)^(-1)(c)=a
 

1. What is an inverse composite proof?

An inverse composite proof is a type of mathematical proof used to show that a statement is true by proving its contrapositive. It involves starting with the opposite of the statement and working backwards to show that it leads to a contradiction or an already proven statement.

2. When is an inverse composite proof used?

Inverse composite proofs are commonly used in algebra, number theory, and other areas of mathematics where logical reasoning is necessary. They are particularly useful when a direct proof or proof by contradiction is difficult to construct.

3. How is an inverse composite proof structured?

An inverse composite proof typically follows a three-step structure: (1) assume the opposite of the statement is true, (2) use logical reasoning to show that this leads to a contradiction or an already proven statement, and (3) conclude that the original statement must be true. This structure is similar to that of a proof by contradiction.

4. What are the advantages of using an inverse composite proof?

One of the main advantages of using an inverse composite proof is that it allows for a more streamlined and logical proof. It also forces the proof writer to think critically and creatively, as they must work backwards to find a contradiction or already proven statement.

5. Are there any limitations to using an inverse composite proof?

While inverse composite proofs can be powerful tools, they may not always be the best approach to proving a statement. In some cases, a direct proof or proof by contradiction may be more straightforward and efficient. Additionally, inverse composite proofs require a strong understanding of logical reasoning and can be more difficult to construct than other types of proofs.

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