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: $b \implies a$

Sufficient: $a \implies b$

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)