Answer :

It is given that f: {1, 2, 3} {a, b, c} given by

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


So, if we define g:


{a,b,c} {1, 2, 3} as


g(a) = 1, g(b) = 2, g(c) = 3, then we get:


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


(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


Therefore, gof = IX and fog = IY, where X = {1, 2, 3} and Y = {a, b, c}


Thus, the inverse of f exists and f-1 = g.


Then, f-1: {a, b, c} {1, 2, 3} is given by


f-1(a) = 1, f-1(b) = 2, f-1(c) = 3


Let us now find the inverse of f-1,


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


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


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


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


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


And,


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


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


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


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


The inverse of g exists and g-1 = h


(f-1)-1 = h


h = f


(f-1)-1 = f


Rate this question :

How useful is this solution?
We strive to provide quality solutions. Please rate us to serve you better.
Try our Mini CourseMaster Important Topics in 7 DaysLearn from IITians, NITians, Doctors & Academic Experts
Dedicated counsellor for each student
24X7 Doubt Resolution
Daily Report Card
Detailed Performance Evaluation
caricature
view all courses
RELATED QUESTIONS :

| Let * be a binaMathematics - Board Papers

Find the idMathematics - Board Papers

Let f : A Mathematics - Exemplar

Show that the binMathematics - Board Papers

Determine whetherRD Sharma - Volume 1

Fill in theMathematics - Exemplar