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

Answer 1

 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