Let A, B &amp; C sets, and left f : A → B and g : B → C be functions. Suppose that f and g have inverses. Prove that g ∘ f has an inverse, and that ( g ∘ f ) − 1 = f − 1 ∘ g − 1 .Assuming that f and g have reverse, f − 1 = h and g − 1 = s with h : B → A, s : C → B.from that above i infer that the inverse of ( g ∘ f ) is ( s ∘ g ) : C → A that is g − 1 ∘ f − 1 = ( g ∘ f ) − 1 ; Hence for proof of ( g ∘ f ) − 1 = f − 1 ∘ g − 1 , proceed as before, only swapping functions, right?

Question

Let   A,   B &amp;amp;   C sets, and left   f  :  A  →  B and   g  :  B  →  C be functions. Suppose that   f and   g have inverses. Prove that   g  ∘  f has an inverse, and that   (  g  ∘  f      )          −      1        =      f          −      1        ∘      g          −      1      .Assuming that f and g have reverse,       f          −      1        =  h and       g          −      1        =  s with   h  :  B  →  A,   s  :  C  →  B.from that above i infer that the inverse of   (  g  ∘  f  ) is   (  s  ∘  g  )  :  C  →  A that is       g          −      1        ∘      f          −      1        =  (  g  ∘  f      )          −      1      ; Hence for proof of   (  g  ∘  f      )          −      1        =      f          −      1        ∘      g          −      1      , proceed as before, only swapping functions, right?

Cristian Delacruz · Accepted Answer

Here is the correct proof.  (  g  ∘  f  )  ∘  (      f          −      1        ∘      g          −      1        )  =  (  g  ∘  (  f  ∘      f          −      1        )  )  ∘      g          −      1        =  (  g  ∘  i      d    B    )  ∘      g          −      1        =  g  ∘      g          −      1        =  i      d    A  and  (      f          −      1        ∘      g          −      1        )  ∘  (  g  ∘  f  )  =       same computations     =  i      d    C  so   (      f          −      1        ∘      g          −      1        ) and   (  g  ∘  f  ) are inverses each other

Let A, B & C sets, and left f:A->B and g:B->C be functions. Suppose that f and g have inverses. Prove that g∘f has an inverse, and that (g∘f)−1=f^−1∘g^−1.

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