1. The problem statement, all variables and given/known data Proof Let F:A→B and g: B→C be functions. Suppose that g°f is injective. Prove that f is injective. 2. Relevant equations 3. The attempt at a solution Let x,y ∈ A, and suppose g°f (x) = g°f(y) and x = y. Suppose g: B→C was not injective. then f(g(x)), if g(x) is some element within B, will be at least two numbers which would no longer make equation injective. If g:B→C was injective then f(c) = f(c') iff c = c'. Suppose f:A→B is not injective. Therefore if f(a') = f(a), a' = a does not need to be equal. Therefore g°f will no longer be injective because a certain element within A can equal to multiple elements within C. Therefore F: A→B must be injective. Tell me why my reasoning does not hold. My teacher sent me this reply. I don't know why my logic doesn't work. "This is *way* too complicated. So much so, that it's not worth me checking whether your reasoning holds in the end (although, I did notice some problems with that, too.) Follow the template! To show f is injective, let a, a' be in A and suppose f(a) = f(a'). Your job is to use the hypotheses to show that a = a'. It should not take more than a few short sentences."