Q. 104.0( 22 Votes )
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) = 1Y(y) = fog2(y). Use one-one ness of f).
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.
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