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)

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)