Question

Problem 6.8. Let X = {1, 2, 3}, Y = {a, b, c, d, e}.

(a) Let f : X → Y be a function, given by f(1) = a, f(2) = b, f(3) = c. Prove there exists a function g : Y → X such that g ◦f = id X . Is g the inverse function to f? (Hint: define g on f(X) to make g ◦ f = id X . Then define g on Y − f(X). Does it matter how you define g on Y − f(X)?)

(b) Is it true that such function g exists for f(1) = a, f(2) = f(3) = c? Justify your answer.

Answer 1

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X through ' . Thia does not diffen the Yeult So if defined g(c)3 9 (6) - 2, You may take 9(e) 2 9(d) = 9(e) = 2 9)=9e)= 3 then, f = id X oY any efher choice you want for d Hence, there exista and function that xuch 1 (Preved) we knaw function have its inverse if amd anby if ana only if it in bijective e, if it in one to - one amd onto A function called for bEB, there exista ang that aE A hla) b uch But for eEY element th ere no in f fx) = Auch that e () a f2) = b t(3)= C f in Hence, not on to Hence, of inverse han no Hence, the imverse of f not

possib le, uepase function Much exiats for fl) =a, f)f(3) = c / must katinfy Then, gaf(2) & got (3) 2 9ito=2, &f(= [from ie, f1 3 9(a)=1, ae) 2 But fact that the contradicts this finetion Because function to two camnot element map Cne ele ments different here, amd g (e)= 2 9 (c) So, our arrumption Was Hence, No uch funetion 3' exista for f)=a, t(2) = f(3)