1. The problem statement, all variables and given/known data If A and B are sets, prove that a subset [itex]\Gamma\subset A X B[/itex] is the graph of some function from A to B if and only if the first projection [itex]\rho: \Gamma\rightarrow A[/itex] is a bijection. 2. Relevant equations 3. The attempt at a solution I first thought that i should define bijection by saying that a a bijection exists when there is both injection and surjection. If [itex]\rho: \Gamma\rightarrow A[/itex] is not a bijection then it is either 1)not surjective 2)not injective 3)both 1) and 2) So, I thought that i should prove that [itex]\Gamma[/itex] is not the graph of some function A -> B when the first projection is not bijective by showing the non-surjective and non-injective cases separately. And then prove that it is, in fact, [itex]\Gamma[/itex] is the graph of the function when the first projection is bijective. So, i have to show the cases for when the first projection is not injective, when it's not surjective, and then when it is bijective. But, how do i prove that the [itex]\Gamma[/itex] is either the graph of the function or not? in any of the 3 cases? I just don't know where to start. Is this the right approach or is there a shorter way to do this?