Ex 1.3, 10 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). Let f: X → Y be an invertible function. If g is an inverse of f, then for all y ∈ Y fog(y) = IY Let g1 & g2 be two inverses of f Then, for all y ∈ Y, fog1(y) = IY & fog2(y) = IY ⇒ fog1(y) = fog2(y) f (g1(y)) = f (g2(y)) Since f is invertible, f is one-one. So, if f(x1) = f(x2), then x1 = x2 Since f (g1(y)) = f (g2(y)), ∴ g1(y) = g2(y) ⇒ g1 = g2 Hence, f has a unique inverse.

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.

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