Surjective functions

  1. 1. The problem statement, all variables and given/known data
    i) Show explicitly that any non-injective function with a right inverse has another right inverse
    ii) Give an example of a function which has (at least) two distinct left inverses

    2. Relevant equations
    i) I don't believe there are any.
    ii) "

    3. The attempt at a solution
    i) Since it says "explicitly" I doubt that my attempt would count as a solution even if it were correct, but here it is;

    Since the function is non-injective and has a right inverse, it must therefore be surjective as it cannot be bijective, and in order for this function to be surjective it must have at least one more right inverse.

    I also tried showing first that f(x) = x^3 - x is non-injective and then showing that the values x = -1,0,1 map to f(x) = 0 that I would be showing this is true but I am not sure if it is correct.

    ii) For this I am really not sure if there is an easier way to do it apart from trial and error but I understand the theory behind it.

    Thanks in advance,
    SD123
     
  2. jcsd
  3. HallsofIvy

    HallsofIvy 40,308
    Staff Emeritus
    Science Advisor

    Try something as simple as this: f:{a, b, c}--> {x, y} defined by f(a)= x,f(b)= x, f(c)= y. g:{x, y}-->{a, b, c}, defined by g(x)= a, g(y)= c is a right inverse.

    Think about that example to prove (i). However, note that an example is NOT a general proof.
     
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