Is This Proof Really a Proof? Examining the Fundamentals of Mathematical Proofs

[tronter]

I figured it out.

 

[jhicks]

If x is any real number? I'd choose a number more interesting than x=\pi. Remember that the claim employs "either/or", so if both \pi - x and x+\pi are irrational the claim is false.

 

[tronter]

So the proof is incorrect because the statement is false? How do you know its an exclusive or?

 

[jhicks]

Yes. Give a counterexample to disprove the claim. It's an exclusive or because that's the language you use when it's an exclusive or. EITHER x-pi is irrational OR x+pi is irrational is what the claim is for any real x, so if they're both irrational then that isn't "either", that's "both".

 

[tronter]

If you changed it to either A... or...B, or both, then the statement and proof would be true?

 

[jhicks]

well strictly speaking just dropping "either" would work, but nobody ever uses "or" to mean logical or. But yes what you say is right.

Edit: Yeah Cristo is right I had totally forgotten about any "proof" by the time I wrote this.

 

[cristo]

The proof is incorrect on a more fundamental level: namely, it is not a proof. It gives one example, but it is claimed that this holds for any x, hence a proof must show this. Remember, you cannot prove a general case by one example.