refer to 14667

Answers (1)

It is given that
f : \left \{ 1,2,3 \right \}\rightarrow \left \{ a,b,c \right \}

f(1) = a, f(2) = b \ and \ f(3) = c

Now,, lets define a function g :
\left \{ a,b,c \right \}\rightarrow \left \{ 1,2,3 \right \}  such that

g(a) = 1, g(b) = 2 \ and \ g(c) = 3

Now,

(fog)(a) = f(g(a)) = f(1) = a
Similarly,

(fog)(b) = f(g(b)) = f(2) = b

(fog)(c) = f(g(c)) = f(3) = c

And

(gof)(1) = g(f(1)) = g(a) = 1

(gof)(2) = g(f(2)) = g(b) = 2

(gof)(3) = g(f(3)) = g(c) = 3

Hence, gof = I_X andfog = I_Y, where X = \left \{ 1,2,3 \right \} andY = \left \{ a,b,c \right \}

Therefore, the inverse of f exists and f^{-1} = g

Now, 
f^{-1} : \left \{ a,b,c \right \}\rightarrow \left \{ 1,2,3 \right \}  is given by

f^{-1}(a) = 1, f^{-1}(b) = 2 \ and \ f^{-1}(c) = 3

Now, we need to  find the inverse of  f^{-1},

Therefore, lets defineh: \left \{ 1,2,3 \right \}\rightarrow \left \{ a,b,c \right \}  such that

h(1) = a, h(2) = b \ and \ h(3) = c  

Now,

(goh)(1) = g(h(1)) = g(a) = 1

(goh)(2) = g(h(2)) = g(b) = 2

(goh)(3) = g(h(3)) = g(c) = 3

Similarly,

(hog)(a) = h(g(a)) = h(1) = a

(hog)(b) = h(g(b)) = h(2) = b

(hog)(c) = h(g(c)) = h(3) = c

Hence,  goh = I_X and hog = I_Y, whereX = \left \{ 1,2,3 \right \} andY = \left \{ a,b,c \right \}

Therefore,  inverse of  g^{-1} = (f^{-1})^{-1}  exists and  g^{-1} = (f^{-1})^{-1} = h

\Rightarrow h = f

Therefore,  (f^{-1})^{-1} = f

Hence proved

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