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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−1and show that (gof)−1 = f −1o g−1
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Solution
f = { ( 1, a ) . (2, b) , (c , 3 ) } and g = {(a , apple) , (b , ball) , (c , cat)} Clearly , f and g are bijections.
So, f and g are invertible.
Now,
f -1 = {(a ,1) , (b , 2) , (3,c)} and g-1 = {(apple, a ) , (ball ,b), (cat , c)}
So, f-1 o g-1= {apple , 1} , (ball,2), (cat , 3 )} ......... (1)
f : {1,2,3,} → {a,b,c} and g : {a,b,c} → {apple , ball , cat}
So, gof : {1.2.3} → {apple , ball, cat}
⇒ (gof) (1) =g (f(1)) = g (a) = apple
(gof) (2) = g (f (2)) = g (b) = ball,
and (gof) (3) = g (f(3)) = g (c) cat
∴ gof = {(1 . apple) ,(2, ball) , (3 , cat)}
Clearly , gof is a bijection.
So, gof is invertible.
(gof)-1 = {(apple , 1), (ball,2),(cat , 3)} ....... (2)
Form (1) and (2) , we get :
(gof)-1 = f-1 o g -1
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