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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).
Solution
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.
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