Question

Let f :X$\to$ 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) = IY (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 ).
Then, for all y ∈Y, we have: fog1 (y)=Iy (y)=fog2 (y)
=>f(g1 (y))=f(g2 (y))
g1 (y)=g2 (y) [f is invertible =>f is one-one
=>g1 = g2 [g is one-one].

Hence, f has a unique inverse.