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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) = IY(y) = fog2(y). Use one-one ness of f).
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Solution
Let f: X → Y be an invertible function.
Also, suppose f has two inverses (say g1 and g2).
Then, for all y ∈ Y, we have:
fog1(y) = Iy(y) = fog2(y)
⇒ f(g1(y)) = f(g2(y)) ...[f is invertible ⇒ f is one-one]
⇒ g1 = g2 ...[g is one-one]
Hence, f has a unique inverse.
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