please fill in EVERY SINGLE blank. Explain as much as possible

(12) Correctness. If your proof is not correct, this is where you'll get docked. You need to (1) precisely define the pigeons (or set A), (1) precisely define the pigeonholes (or set B), (1) precisely define the function f : A - B between them, (3) briefly explain why f is well-defined, using the 3 properties of a function (3) explain why we are able to apply the PHP, (1) correctly apply the PHP using your specific, precise definition of f and A, (2) explain how the result of the PHP achieves the desired result of this problem. (4) Communication. We need a mix of notation and intuition in the "column" format (1) Use the "column format": each row is a precise mathematical statement on the left, and the reason that statement is true on the right (or on the next line) (1) The mathematical statements must be precise and mostly (if not all) algebraic. (1) The reasons must be taken from the approved list and should be written out in full. (1) Provide enough rows / steps so that we can follow your proof, but do not have redundant or useless steps that might make the proof unnecessarily confusing