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Consider f:{1,2,3}{a,b,c} given by f(1)=a,f(2)=b and f(3)=c. Find f1 and show that (f1)1=f.
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Solution:

If we define g:{a,b,c}{1,2,3} as g(a)=1,g(b)=2,g(c)=3, then we have:

(fg)(a)=f(g(a))=f(1)=a(fg)(b)=f(g(b))=f(2)=b(fg)(c)=f(g(c))=f(3)=c

And,

(gf)(1)=g(f(1))=g(a)=1(gf)(2)=g(f(2))=g(b)=2(gf)(3)=g(f(3))=g(c)=3

gf=IX

and

fg=IY

where

X={1,2,3}

and

Y={a,b,c}.

Thus, the inverse of f exists and f1=g.

f1:{a,b,c}{1,2,3} is given by,

f1(a)=1,f1(b)=2,f1(c)=3

Let us now find the inverse of f1 i.e., find the inverse of g.

If we define h:{1,2,3}{a,b,c} as

h(1)=a,h(2)=b,h(3)=c, then we have:

(gh)(1)=g(h(1))=g(a)=1(gh)(2)=g(h(2))=g(b)=2(gh)(3)=g(h(3))=g(c)=3

And,

(hg)(a)=h(g(a))=h(1)=a(hg)(b)=h(g(b))=h(2)=b(hg)(c)=h(g(c))=h(3)=c

goh=IX and hog=Iγ, where X={1,2,3} and Y={a,b,c}.

Thus, the inverse of g exists and g1=h(f1)1=h.

It can be noted that h=f.

Hence,

(f1)1=f.

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