Question

Let f: X → Y be an invertible function. Show that f has unique inverse. (Hint: suppose g₁ and g₂ are two inverses of f. Then for all y ∈ Y, fog₁(y) = I_Y(y) = fog₂(y). Use one-one ness of f).

Answer 1

Let f: X → Y be an invertible function.

Also, suppose f has two inverses (say `g_1` and `g_2)`_).

Then, for all y ∈Y, we have:

`fog_1(y) = I_y (y) = fog_2 (y)`

`=> f(g_1(y)) = f(g_2(y))`             [f is invertible => f is one-one]

`=> g_1 = g_2`      [g is one- one]

Hence, f has a unique inverse.