Prove If f X Y is 11 and onto then there is a function g YX

Prove: If f: X-> Y is 1-1 and onto, then there is a function g: Y->X that is 1-1 and onto and such that g(f(x)) = x for all x in X.

Solution

Let f XY be a one-to one and onto function and let g = f^-1 : YX be its inverse. We shall show first that f^-1 is one-to one. Let us assume to the contrary. Let x₁ and x₂ Y so that f^-1 (x₁) = f^-1(x₂). Then since f^-1 is the inverse of f, we have x₁ = f ( f^-1(x₂)) i.e. x₁ = x_2. Thus f^-1 is one-to-one.

Now, we shall prove that f^-1 is also onto. Let x be an arbitrary element of X. We have to show that f^-1 maps some element of Y to x. Now x = f^-1 (f(x)) . Since f is onto, it maps x in X to some element ( say) y of Y . Then x = f^-1 (f(x)) = f^-1(y) for some y in Y . Thus f^-1 is also onto. Thus the inverse function g = f^-1 is both 1-1 and onto.