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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. Showthat f, g and gof are invertible. Find f−1, g−1 and gof−1andshow that (gof)−1 = f −1o g−1



Answer -

Given f = {(1, a), (2,b), (c , 3)} and g = {(a , apple) , (b , ball) , (c , cat)} Clearly , f and gare bijections.

So, f and g areinvertible. 

Now,

-1 ={(a ,1) , (b , 2) , (3,c)} and g-1 = {(apple, a), (ball , b),(cat , c)}

So, f-1 og-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 abijection.

So, gof isinvertible. 

(gof)-1 = {(apple,1), (ball, 2), (cat, 3)}……. (2)

Form (1) and (2), weget

(gof)-1 =f-1 o g -1

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