Proofs and non-Boolean Functions, disproofs, and false proofs!

Non-Boolean Functions

In class, we discussed about non-boolean functions such as the floor function which takes in a number and returns the largest integer that is less than or equal to that number. The non-boolean functions can be used to set up more interesting predicates for evaluation. For example, prove that n² + 3n + 127 is a prime number for all natural numbers n. Ok, lets take natural number 0, yes, 127 is a prime number. What about 1? If n = 1, then it equals to 131, yes that is a prime. 2? No...2 is not prime. So the proposition is incorrect. So for non-boolean functions, some input values would be ok, while others are not. The main takeaway is that just because a predicate seems to be true for some values, doesn't mean the whole proposition is true.

Disproving

Now, how do we disprove something? For example, how do we disprove the statement 2+2=5? We just need to prove that 2+2 does not equal to 5. So to disprove any statement S, we need to prove that not S is true. This is very useful, because we would not always get to prove true propositions, we would need to know how to disprove something that is false. The most important thing to notice is that the proposition is false, then we can negate it and start disproving. If we fail to notice that a proposition is false, we can try to prove it, but it wouldn't go anywhere. It is a very bad thing to prove something False to be True.

False Proofs

I think this is the funniest topic in this course. We must remember that the real world exists outside of the realm of mathematics, and there are many types of proofs out there, some of which do not actually prove anything! I think the most prevalent false proof out there is the proof by gut feeling. Our intuition is good for rough estimations and predictions based on bad evidence, which was useful in the wild, when we needed to escape predators lurking in the bushes, but not good for establishing truths.