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) = 1_Y(y) = fog₂(y). Use one-one ness of f).

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) = 1_Y(y) = fog₂(y). Use one-one ness of f).

Philoid · Accepted Answer

It is given that f: X → Y be an invertible function.

Also, suppose f has two inverse

Then, for all y ϵ Y, we get:

fog₁(y) = I₁ (y) = fog₂(y)

= > f(g₁(y)) = f(g₂(y))

= > g₁(y) = g₂(y)

= > g₁ = g₂

Therefore, f has a unique inverse.