Proof of Theorem: Composite Function Inverse

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The discussion centers on proving the theorem that states if functions f and g are bijective, then the inverse of their composition (g o f) equals the composition of their inverses (inverse of f o inverse of g). A user requests assistance with the proof, prompting suggestions to define f(x) = y and g(y) = z. The hint encourages computing (g o f) o (g o f)^-1 and (g o f)^-1 o (g o f) to demonstrate that both yield the identity function. This approach is aimed at establishing the validity of the theorem through direct computation. The conversation emphasizes the importance of understanding function composition and inverses in the context of bijective functions.
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i really need to see the proof of this theorem:

if f and g are bijective then the inverse of (g o f) = inverse of f o inverse of g
 
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i hope you will help me..
 
what have you tried?
 
Hint: define f(x) = y and g(y) = z.
 
Just compute (g o f) o (g o f)^-1 and (g o f)^-1 o (g o f) and see that they both give you the identity.
 

Thread 'Hypothesis testing: Defining H0, HA hypotheses so that ( H_A)_A' makes sense'

The standard _A " operator" maps a Null Hypothesis Ho into a decision set { Do not reject:=1 and reject :=0}. In this sense ( HA)_A , makes no sense. Since H0, HA aren't exhaustive, can we find an alternative operator, _A' , so that ( H_A)_A' makes sense? Isn't Pearson Neyman related to this? Hope I'm making sense. Edit: I was motivated by a superficial similarity of the idea with double transposition of matrices M, with ## (M^{T})^{T}=M##, and just wanted to see if it made sense to talk...
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