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Question

Consider f : {1, 2, 3} → {a, b, c} and g : {a, b, c} → {apple, ball, cat} defined as f (1) = a, f (2) = b, f (3) = c, g (a) = apple, g (b) = ball and g (c) = cat. Show that f, g and gof are invertible. Find f−1, g−1 and gof−1 and show that (gof)−1 = f 1o g−1.

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Solution

f=1, a, 2, b, 3, c and g=a, apple, b, ball, c, catClearly, f and g are bijections.So, f and g are invertible.Now,f-1=a, 1, b, 2, c, 3 and g-1=apple, a, ball, b, cat, cSo, f-1o g-1=apple, 1, ball, 2, cat, 3 ...1f:1, 2, 3a, b, c and g:a, b, capple, ball, catSo, gof:1, 2, 3apple, ball, catgof 1=g f 1=g a=applegof 2=g f 2=g b=ball,and gof 3=g f 3=g c=catgof =1, apple, 2, ball, 3, catClearly, gofis a bijection.So, gof is invertible.gof-1=apple, 1, ball, 2, cat, 3 ...2From 1 and 2, we get:gof-1=f-1o g-1

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