Can you ever disprove a proof or prove something in more than one way

Question

This seems to be a very simple question but surely because there is an infinite way of writing a=1 and b=2 in so many different forms, can you ever disprove a proof or prove something in more than one way. Thanks

Answer 1

You cannot disprove a proof. Instead, what you are doing is proving that said proof is not a proof.

And yes, there are generally many ways to prove something. At the simplest, most things can be proven directly, by induction, or by contradiction.

Answer 2

if "disprove a proof" means "find out that a statement is false", there are many ways to do it too. The first ones I can think of are:

• to find an explicit counterexample to the statement. If you state "there are no solution to the equation $|{3^m-2^n}|=1; m,n\ge 2$", I may say $3^2-2^3=1$.
• to find a contradiction, that is, show that if the statement were true you would derive a result which is certainly false.
• a variant of contradiction is to show that there cannot exist a minimum or a maximum value for a finite set; the Euclidean proof that the prime numbers are infinite works in this way.
• still another variant is to show that a property which is necessary is not achieved. Euler showed that all nodes in the graph for a reentrant path must have an even number of edges exiting from them; the graph for Königsberg walk had four nodes with an odd number of edges, so it could not be reentrant.