Q.10 Let f : X \rightarrow Ybe an invertible function. Show that f has a unique inverse.
(Hint: suppose g_1 and g_2 are two inverses of f. Then for all y \in Y,
fog_1 (y) = I_Y (y) = fog_2 (y). Use one-one ness of f).

Answers (1)

Let f : X \rightarrow Ybe an invertible function

Also, suppose f has two inverse g_1 and g_2

For y \in Y, we have

     fog_1(y) = I_y(y)=fog_2(y)

 \Rightarrow     f(g_1(y))=f(g_2(y))  

\Rightarrow             g_1(y)=g_2(y)                     [f is invertible implies f is one - one]

\Rightarrow                     g_1=g_2                        [g is one-one]

Thus,f has a unique inverse.

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