User:KarlFrei

House Rules
''The following are the house rules as I currently understand them. I had to piece them together bit by bit as they are not written down anywhere that I could find. As such, there may be things I have misunderstood; I welcome corrections.''

You will have seen our house style. But did you know that we also have house rules? These are probably the most important rules that new users (particularly mathematicians, who should tread very lightly on this website) should know about, but to keep things interesting, we do not mention them anywhere in Help:Editing or our house style, and we certainly don't tell new users about them when they join. Where would be the fun in that? There is plenty of time to tell them when they start not following these rules (for some strange reason).

Here they are, in order of importance.


 * 1) There is to be no deletion of any material. Ever. (You can tell because the pages Help:Editing and our house style conspicuously do NOT contain the word "delete" or "deleting". How much more obvious could we have made this policy?)
 * 2) There is to be no change in any material on this website UNLESS
 * 3) it is demonstrably wrong (and even then, it might be kept for historical interest)
 * 4) it is not in house style
 * 5) it is an expansion of existing material, by adding links to related concepts or definitions, adding missing steps in proofs, adding explanations for steps in proofs, etc. etc. ...
 * 6) (As a logical consequence of rules 1 and 2 above:) Editing on this website means expanding and expanding only. Even though we have a general disclaimer stating that your text may be "altered" and even "edited mercilessly", that is just a bit of a joke we put in for laughs. If you write anything on this website, and it is mathematically correct and in the proper style, you may rest assured that it will still be there in a hundred years. Never mind if your proof is circuitous or hard to follow or understandable only by you: if we cannot show it is actually wrong, it will stay. Go for it! On the other hand, if you happen to see any step in a proof that can be simplified, then unfortunately the only way to get the simpler step on ProofWiki is to write a completely separate proof with this change, no matter how small and easy the change or how big the proof. Do not under any circumstances attempt to "improve" any proof by editing it. (Mathematicians beware!) If you come here hoping to possibly make some proofs easier, you can do that by adding a new version, but the version which you think is bad MUST stay on the website next to it, because we say so.
 * 7) Regarding the house style. This is another one of our little jokes. Even though that page says "House style", these are actually also RULES, and they are very strict and incredibly important. Following them is almost as important, if not more important, than actually contributing mathematics to this website. Ignore them and you will keep getting reminders until you either start obeying the rules (because they must be obeyed) or give up in disgust or possibly get blocked. As for questioning any rules, you are free to try it, and good luck!
 * 8) Introducing a general "Notes" section is discouraged. Again, you could have seen this by noticing that there is no section "Notes" on the help pages, only more specific sections like "Historical notes". Must we spell out everything?
 * 9) Discussing this list of rules is discouraged. See here.

To summarize: even though the main page of ProofWiki has collaboration as a primary goal, this should not be taken to mean that you can just start and change text as you would on that nasty Wikipedia. Remember the mantra: editing means expanding or putting articles into house style.

Proofs by Contraposition
The below was collected in response to the statement made on my talk page that "Whether you prefer it to be a proof by contraposition or not, does not mean it is a proof by contraposition until it is turned into it."

Source 1
There is a useful rule of thumb, when you have a proof by contradiction, to see whether it is "really" a proof by contrapositive.

In a proof of by contrapositive, you prove P→Q by assuming ¬Q and reasoning until you obtain ¬P.

In a "genuine" proof by contradiction, you assume both P and ¬Q, and deduce some other contradiction R∧¬R.

So, at then end of your proof, ask yourself: Is the "contradiction" just that I have deduced ¬P, when the implication was P→Q? Did I never use P as an assumption? If both answers are "yes" then your proof is a proof by contraposition, and you can rephrase it in that way.

For example, here is a proof by "contradiction":

Proposition: Assume A⊆B. If x∉B then x∉A.

Proof. We proceed by contradiction. Assume x∉B and x∈A. Then, since A⊆B, we have x∈B. This is a contradiction, so the proof is complete.

That proof can be directly rephrased into a proof by contrapositive:

Proposition: Assume A⊆B. If x∉B then x∉A.

Proof. We proceed by contraposition. Assume x∈A. Then, since A⊆B, we have x∈B. This is what we wanted to prove, so the proof is complete.

Proof by contradiction can be applied to a much broader class of statements than proof by contraposition, which only works for implications. But there are proofs of implications by contradiction that cannot be directly rephrased into proofs by contraposition.

Proposition: If x is a multiple of 6 then x is a multiple of 2.

Proof. We proceed by contradiction. Let xbe a number that is a multiple of 6 but not a multiple of 2. Then x=6y for some y. We can rewrite this equation as 1⋅x=2⋅(3y). Because the right hand side is a multiple of 2, so is the left hand side. Then, because 2 is prime, and 1⋅x is a multiple of 2, either x is a multiple of 2 or 1 is a multiple of 2. Since we have assumed that x is not a multiple of 2, we see that 1 must be a multiple of 2. But that is impossible: we know 1 is not a multiple of 2. So we have a contradiction: 1 is a multiple of 2 and 1 is not a multiple of 2. The proof is complete.

Of course that proposition can be proved directly as well: the point is just that the proof given is genuinely a proof by contradiction, rather than a proof by contraposition. The key benefit of proof by contradiction is that you can stop when you find any contradiction, not only a contradiction directly involving the hypotheses.



It's not the same.

If P and Q are statements about instances that (a priori independently) are true for some instances and false for others then proving P⇒Q is the same as proving the contrapositive ¬Q ⇒¬P. Both mean the same thing: The set of instances for which P is true is contained in the set of instances where Q is true.

Proving a statement A by contradiction is something else: You add ¬A to your list of axioms, and using the rules of logic arrive at a contradiction, e.g., at 1=0. Then you say: My axiom system was fine before adding ¬A. Since this addition has spoiled it, in reality A has to be true.



Source 2
When coming to prove P⇒Q, we can either:
 * 1)     Prove directly, that is assume P and show Q;
 * 2)     Prove by contradiction, that is assume P and ¬Q and derive contradiction; or
 * 3)     Prove the contrapositive, that is assume ¬Q and show ¬P.