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) = 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 ). 
Then, for all y ∈Y, we have: fog₁ (y)=I_y (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.