To start a proof, it is important to first understand the statement or problem that you are trying to prove. Then, you can begin by stating your assumptions and defining any necessary terms. From there, you can use logical steps and previous theorems to build your proof.
Axioms are self-evident statements that are accepted as true without needing to be proven. In a proof, you can use any axioms that are applicable to the problem at hand. It is important to carefully consider which axioms are relevant and necessary for your proof.
A valid proof follows a logical and coherent sequence of steps that lead to the conclusion you are trying to prove. It is important to check that each step is justified and that the conclusion logically follows from the given assumptions and axioms. Additionally, you can seek feedback from others to ensure the validity of your proof.
A counterexample is an example that disproves a statement. If you encounter a counterexample in your proof, it means that the statement is not universally true. You can either revise your proof to account for the counterexample or modify the statement to make it more specific.
Practice and feedback are key to improving your proof writing skills. It is important to regularly work on challenging problems and to seek feedback from others, such as professors or peers. Additionally, familiarizing yourself with different proof techniques and strategies can also help improve your skills.