Question -
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.┬а
Now,
f┬а-1┬а={(a ,1) , (b , 2) , (3,c)} and g-1┬а= {(apple, a), (ball , b),(cat , c)}
So, f-1┬аog-1= {apple, 1), (ball, 2), (cat, 3)}тАжтАжтАж (1)
f: {1,2,3,}┬атЖТ {a,b, c} and g: {a, b, c}┬атЖТ {apple, ball, cat}
So, gof: {1, 2,3}┬атЖТ {apple, ball, cat}
тЗТ (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┬а
тИ┤ gof = {(1, apple),(2, ball), (3, cat)}
Clearly, gof is abijection.
So, gof isinvertible.┬а
(gof)-1┬а=┬а{(apple,1), (ball, 2), (cat, 3)}тАжтАж. (2)
Form (1) and (2), weget
(gof)-1┬а=f-1┬аo g┬а-1