Mark44 said:Something you asserted early on is not true - that f(g(x)) = g(f(x)). I stopped reading after that.
You would have been doing a proof by contradiction if you said you were doing a proof by contradiction. The way you've written it, you're just asserting (without justification) something that is not even true.Sylvester Sly said:I was doing proof by contradiction. In order to prove f(g(x)) =/= g(f(x)) i started off by letting f(g(x)) = g(f(x)) and working from there.
I assumed that was a typos for "g(f(x))= f(g(x))= cos(sin(x))". But, in any case, if the problem was to determine which was larger, I don't see what a proof by contradiction that they are not equal would accomplish.Mark44 said:If you're doing a proof by contradiction, you don't start with "therefore ..." - You start by assuming the thing you want to contradict.
Also, in the same line you have "Therefore g(f(x)) = f(g(x)) cos(sin(x))". It looks like you omitted part of what you wanted to say.
A proof is a logical and rigorous method used to establish the truth or validity of a statement or theorem. It is used to convince others of the validity of a mathematical or scientific argument.
A proof is considered correct if it follows the rules of logic and is free of errors. It must also be based on accepted axioms and definitions, and it should be able to withstand scrutiny and be reproducible.
A proof is considered sufficient if it provides enough evidence or reasoning to support the conclusion of a statement or theorem. This means that it must be convincing and leave no doubts about the validity of the argument.
No, a proof can never be considered absolute or infallible. It is always possible that new evidence or a new understanding of the subject can emerge, leading to a different conclusion or a need for a revised proof.
Some common mistakes to avoid when constructing a proof include using circular reasoning, making assumptions without justification, and using incorrect or incomplete logic. It is also important to ensure that all steps are clearly and logically presented, and that all necessary information is included in the proof.