User talk:Lord Farin/Proof Structures

Thank you for this! --GFauxPas 15:49, 4 November 2011 (CDT)

Notes to Set Equality Proofs
A few points:

a) Might want to add a link to the justification for $\complement \left({A}\right) \subseteq \complement \left({B}\right) \iff B \subseteq A$ so as to ensure rigor.


 * I'd actually put it something like this:

"From Equality of Sets, we need to show $A \subseteq B$ and $B \subseteq A$.

From Complements Invert Subsets:
 * $\complement \left({A}\right) \subseteq \complement \left({B}\right) \iff B \subseteq A$

Thus it suffices to prove $A \subseteq B$ and $\complement \left({A}\right) \subseteq \complement \left({B}\right)$."

Then there is no ambiguity as to what is being demonstrated, and indeed why we need to show $A \subseteq B$ and $B \subseteq A$ in the first place.

(We've had some fuss raised in the past along the lines: "There's no need to put this basic stuff in, because this proof is so abstruse that if someone's reading it they must already know it." I demur.) --prime mover 13:38, 5 November 2011 (CDT)


 * Again, I have to agree; hence I adapted the structures. This site exists to give proofs with maximal rigour, so we should not be afraid to give it (at least, that's my take on it). I have to thank you, for you expand my knowledge of English almost every time I read one of your sentences (being a non-native speaker myself). Interestingly, most of the words I don't know are described on the various dictionary sites as archaic or obsolete ;). --Lord_Farin 18:36, 5 November 2011 (CDT)


 * PML. It has been noted before! I possess a predilection for the utilisation of multisyllabic morphological constructs. --prime mover 18:54, 5 November 2011 (CDT)

b) Note that house style uses $\subseteq$ rather than $\subset$ for possibly-not-proper subset, so as to minimise ambiguity.

c) May be just me but I'm not sure about the "we have" in the wording: IMO the line is best as just "Then $A = B$". I'm okay with "we have" and so on in proofs, if it makes the exposition clearer, but a formal statement of the result requires formal language. --prime mover 06:08, 5 November 2011 (CDT)
 * Good points. I put this up in a hurry, had to leave. Concerning c), I agree; even in proofs I find myself avoiding 'we' and similar terms. Concerning b), I think it is best to always use $\subsetneq$ instead of $\subset$, again to minimise ambiguity.
 * Yeah agreed, I'm prepared to go along with that. The issue hasn't come up very often because the majority of the time when $\subset$ is used, it's $\subseteq$ that's required (i.e. "proper subsets" don't seem to be needed much).
 * Another thing: We might want to consider putting these templates of us in some page (for example in the proof writing guidelines, which are almost empty), together with links to examples of usage. I will also think about creating an extension that allows for copy-paste regions (i.e., teletype formatting without wiki processing), to surpass the annoying necessity to edit the page and then copy. It is likely this will take considerable time though. --Lord_Farin 10:04, 5 November 2011 (CDT)
 * Yeah we could do ... you're actually the first one to raise this issue. If you have a technique that you can implement, then I'd be well happy. --prime mover 13:32, 5 November 2011 (CDT)

Notes on "necessary and sufficient" language
I see a lot of proofs broken up into sections on "necessary" and "sufficient" or similar without first setting up the necessary context. If I wish to prove $a \iff b$, neither direction of proof is inherently one or the other. Some options:

Theorem
$a \iff b$

Proof
$a \implies b$: blah blah

$b \implies a$: blah blah

Theorem
$a \iff b$

Proof
Implication: proof that $a \implies b$

Reverse implication: proof that $b \implies a$

Theorem
$a$ if and only if $b$

Proof
If: Proof that $b \implies a$

Only if: Proof that $a \implies b$

Theorem
$a$ is necessary and sufficient for $b$

Proof
Necessary: proof that $b \implies a$

Sufficient: proof that $a \implies b$

Addendum: my basic point is that the language used in the proof should be grounded in the language used in the theorem. --Dfeuer (talk) 08:32, 8 January 2013 (UTC)


 * "Necessary condition" and "Sufficient condition", when $a \iff b$, are defined here. I use them as section names in the proofs I upload, mainly to keep to consistency. --Anghel (talk) 08:50, 8 January 2013 (UTC)


 * The problem, Anghel, is that while $a \iff b$ means the same thing as "$a$ is necessary and sufficient for $b$", it also means the same thing as "$b$ is necessary and sufficient for $a$". Relying on a ProofWiki-specific convention for which is which is both confusing to people not steeped in PW-ness and fragile in the presence of fallible humans. Establishing context using words is clearer and more reliable, in my view. --Dfeuer (talk) 08:59, 8 January 2013 (UTC)


 * While such would be clearer, in many cases both $a$ and $b$ are very involved expressions which cannot be nicely put into two or three words. Admittedly there is room for improvement (as always) - people may not be familiar with the "Necessary" and "Sufficient" nomenclature. I always thought it was quite obvious that we reasoned from the part mentioned first; am I reasoning too much from my personal presuppositions here? --Lord_Farin (talk) 09:23, 8 January 2013 (UTC)


 * I think so. I agree that forms that repeat everything are often poor choices, but it should generally be easy to rephrase the statement of the theorem to clarify which direction is to be considered necessary and which sufficient. --Dfeuer (talk) 09:30, 8 January 2013 (UTC)


 * Some pages also use the alternative of naming/numbering the propositions, so that the implications can be written out compactly. --Dfeuer (talk) 09:31, 8 January 2013 (UTC)


 * By the way, a bit of history: Previously, the two "Conditions" used to be links to the pages defining these notions. With the advent of the transclusion paradigm for multiple theorems, proofs and definitions, this had to be eliminated to avoid confusion (there are also proofs now that split Nec and Suf, making the titles links as well). I'm not currently up for a rewriting marathon for such a minor point, but I suppose that if we can decide on a good new approach I'll implement it as I tread. --Lord_Farin (talk) 09:41, 8 January 2013 (UTC)
 * The solution might be as simple as a remark on the first line of the proofs of the implications (and usually it's already apparent from this line which implication is being dealt with). We could also use "Forward Implication" and "Reverse Implication". I'm not a fan of "If" and "Only If" because this does not correspond to natural language regarding implications. --Lord_Farin (talk) 09:45, 8 January 2013 (UTC)

I feel like we're talking at cross-purposes. I have no difficulty with the notions of necessary conditions or sufficient conditions. These are utterly standard. What's not standard, and is confusing, is a convention for determining which of two sides of an "iff" or $\iff$ is the condition relative to which the other is to be considered necessary or sufficient. I am quite accustomed to seeing, say, "Necessary and sufficient conditions for blah", followed by a list of necessary conditions for blah and a list of sufficient conditions for blah. I'm not accustomed to the terms popping out of nowhere.

That said, I think "forward implication" and "reverse implication" are nice English translations of $\implies$ and $\impliedby$. --Dfeuer (talk) 09:53, 8 January 2013 (UTC)


 * Since we determined that it is (at least sometimes) unwieldy to adapt the terminology NecSuf to explicate which of the two options has been chosen, I proceeded to naming alternatives and to comment on your suggestions. I get the problem; I was merely trying to resolve it and wanted to inform you on how the construct arose in the first place. --Lord_Farin (talk) 10:00, 8 January 2013 (UTC)


 * My solution has been to re-state the assumption $a$ or $b$ in the very first line of the Necessary/Sufficient Condition paragraph, so people will know what I'm trying to prove - something like Lord_Farin mentioned above. But I do agree that the terms are slightly confusing for newcomers. I'm mainly using these words as titles because they were standard on this wiki before I began writing here - I find that in most theorems, it's hard to shorten $a$ and $b$ to titles. --Anghel (talk) 10:08, 8 January 2013 (UTC)

IMO the very point being raised is completely spurious. What is important is that both sides of the implication get proved. Which is what is neither here nor there. However we resolve this, the prospect of having to completely redo all this again makes me want to puke. --prime mover (talk) 10:56, 8 January 2013 (UTC)

Induction
The framework for induction here is very heavy. Thinking about it, I think the problem is the "inductive hypothesis" section. Unlike the other two sections, it is not itself a proof at all, although the format suggests it to be a lemma like the others. Identifying/naming the inductive hypothesis explicitly may be common in the first few exercises of some texts using it, but it's not actually a logically element of the proof. All you need, logically, to prove that $P(n)$ holds for each natural $n$ is to prove that $P(0)$ holds and that $\forall n \in \N_n: \bigl({ P(n) \implies P(n+1) }\bigr)$. The key to clarity, I believe, is to make sure there can be no confusion over the identities of $P$ and $n$. --Dfeuer (talk) 17:40, 8 March 2013 (UTC)


 * BLARGH. Someone (me) needs to look in some good texts that do a lot of induction and figure out exactly what it is that they do to avoid so much boilerplate. --Dfeuer (talk) 17:54, 8 March 2013 (UTC)


 * Absolutely nothing wrong with the way we present induction proofs. --prime mover (talk) 19:11, 8 March 2013 (UTC)