1. The problem statement, all variables and given/known data Give a genuinely new (i.e. not discussed in class or in the book or in tutorial) example of: two sets X and Y , and two functions f : X →Y and g : Y → X, such that the composition g ◦ f is the identity function 1X : X → X, but neither f nor g are bijective. (Reminders: if f : X→ Y and g : Y → Z are two functions, then the composition g ◦ f is a function from X to Z defined by (g◦f)(x):=g(f(x))forallx∈X.The identity function1X :X→X is defined by1X(x):=x for all x ∈ X.) 2. Relevant equations 3. The attempt at a solution I was just wondering since f and g are inverses of each other, wouldn't they have to be bijective in order for their composition to be the identity function X --> X? In general, I'm just very confused that if you put in a certain input, you would get the same input as an output. Wouldn't the set size of f also be equal to the set size of g?