[OS X TeX] Latex symbol for "define equal"

Tue Aug 21 13:47:26 CEST 2018

The thing is that you can have several different definitions, all equivalent.  Consider the definition "a matrix A \in \mathbb R^{n \times n} is invertiable iff A have a multiplicative inverse". It turns out that this is equivalent to a lot of things, among det(A) !=0, A having full rang, A having n linear independent eigenvectors, Ax=0 only having the trivial solution, Ax=b having a solution (which is unique) for every right-hand-side b, etc.

In these cases it doesn't make much sense to use "if" and lose the importance that all these other statements, which could be taken as definitions. are equivalent. An other example: consider what it means for a complex to be holomorphic. It turns out that holomorphic and analytical is the same for complex functions, but it's not the same on R. You can have a real smooth non-analytical function, so clearly one should be careful to treat those concepts as the same. Which is why you keep "iff" in the definition of holomorphicy, not fooling students that entire and analytical functions are the same.

________________________________
Från: MacOSX-TeX <macosx-tex-bounces at email.esm.psu.edu> för Josep Maria Font <jmfont at ub.edu>
Skickat: den 21 augusti 2018 13:12
Till: TeX on Mac OS X Mailing List
Ämne: Re: [OS X TeX] Latex symbol for "define equal"

If the context does not make it clear whether you are reading a theorem or a definition, something is wrong with this book !

Normally the reader is reading a statement marked as "Definition", or one marked as "Theorem" (Lemma, Proposition, etc.). And in the Preliminaries you can explain your usage of "when" in definitions as meaning "if and only if".

In this way you keep the "if" family, including "if and only if", for real mathematical statements, that is, statements which you can (in theory) check whether they are true or false; notice that one of the main differences between a definition and a theorem is that it does not make sense to say that a definition is true or false!

JMaF

El 21 /08/18, a les 13:02, Markus Klyver <markusklyver at hotmail.com<mailto:markusklyver at hotmail.com>> va escriure:

But how would the reader know if it's a theorem or a definition? Consider the statement "f(x)=1 when x=2". Do we mean f(x)=1 <---> x=2 or just f(x)=1 <--- x=2?

________________________________
Från: MacOSX-TeX <macosx-tex-bounces at email.esm.psu.edu<mailto:macosx-tex-bounces at email.esm.psu.edu>> för Josep Maria Font <jmfont at ub.edu<mailto:jmfont at ub.edu>>
Skickat: den 21 augusti 2018 12:15
Till: TeX on Mac OS X Mailing List
Ämne: Re: [OS X TeX] Latex symbol for "define equal"

That's why I prefer to use "when" for definitions: [...this NEW thing happens...] WHEN [...such-and-such known, understandable condition holds...].

JMaF

> El 21 /08/18, a les 5:40, juan tolosa <juantolo at me.com<mailto:juantolo at me.com>> va escriure:
>
> This is tangential but, since we are at it, what really, really annoys me, is the tendency in some books to define concepts using “if and only if” instead of plain “if”, as in
>
> Definition. A sequence (a_n) converges to a real number p if and only if for every epsilon > 0 there is an N such that (etc.)
>
> (Is it my impression, or there is a growing number of such texts? And when did this nonsense begin?)
>
> One could simply agree that the “if” in definitions is not the same as the “if” in logical statements.
> If one is really fastidious, one could put a statement at the beginning of the book that “if” in definitions can be reworded as, say,
> by “(a_n) converges to p” we mean … (etc)
> Or simply not use “if” at all in definitions.
> What is even more annoying is that when the definition gets really involved, the “if and only if” makes it even worse.
> And, invariably, in a really involved definition you will find that the author(s) abandon their own fastidiousness and just use “if.”
> Question is, why not do it from the very beginning?
>
> Juan
>
>> On Aug 20, 2018, at 7:56 PM, Markus Klyver <markusklyver at hotmail.com<mailto:markusklyver at hotmail.com>> wrote:
>>
>> Really, there's no standardized notation for it. Just be consistent and use the most aesthetically pleasing and practical notation on your context. Maybe := looks ugly will your fonts. Use an other notation instead.
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