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Question -

Consider┬аf: {1, 2, 3} тЖТ {a,┬аb,┬аc} and┬аg:{a,┬аb,┬аc} тЖТ {apple, ball, cat} defined as┬аf┬а(1)=┬аa,┬аf┬а(2) =┬аb,┬аf┬а(3) =┬аc,┬аg┬а(a) =apple,┬аg┬а(b) = ball and┬аg┬а(c) = cat. Showthat┬аf,┬аg┬аand┬аgof┬аare invertible. Find┬аfтИТ1,┬аgтИТ1┬аand┬аgofтИТ1andshow that (gof)тИТ1┬а=┬аf┬атИТ1o┬аgтИТ1



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

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