[OS X TeX] Latex symbol for "define equal"
Ross Moore
ross.moore at mq.edu.au
Sun Aug 26 10:11:49 EDT 2018
Sorry to continue this thread.
On 21/08/2018, at 13:49, "Markus Klyver" <markusklyver at hotmail.com<mailto:markusklyver at hotmail.com>> wrote:
Definitions *are* equivalences.
But most equivalences are certainly not definitions.
So the same symbols should not be used.
(For example, 5 is equivalent to 2 modulo 3. But it would be silly to use 5 as a canonical example of its equivalence class mod 3.)
A definition should be introducing the *words* that are going to be used to refer to a mathematical concept. Frequently a notation to represent the concept will also be presented.
In a sense the definition is necessarily a tautology, hence an equivalence.
But to a student who has not met the terminology already, such a characterization is completely useless.
After further study it becomes apparent that there can be many alternative ways to see that a particular mathematical property holds, as in the det A = 0 example. But these are not alternative definitions, despite the fact that some authors try to say that they are. That they are equivalent *properties* is usually presented as a theorem, and each requires a calculation to demonstrate the equivalence.
Texts that introduce new concepts by means of a definition, without previous motivating examples, are notoriously difficult to learn from. That's why Bourbaki never really gained much traction — if you are not already an expert, just forget it.
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Från: MacOSX-TeX <macosx-tex-bounces at email.esm.psu.edu<mailto:macosx-tex-bounces at email.esm.psu.edu>> för Ross Moore <ross.moore at mq.edu.au<mailto:ross.moore at mq.edu.au>>
Skickat: den 21 augusti 2018 13:21
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Ämne: Re: [OS X TeX] Latex symbol for "define equal"
Hi all.
I agree with using "when", or better "precisely when", in definitions.
Do not use these words in theorems. There are many alternative ways to express what you want.
But didn't this thread start out asking for a symbolic notation?
In that context, equivalence is *not* the word you want for a definition, as equivalence depends on a particular relation, so is weaker than a definition.
There are nuances in mathematical concepts and notations that are not always correctly appreciated by those who work primarily in other (mostly applied) fields.
On 21/08/2018, at 13:03, "Markus Klyver" <markusklyver at hotmail.com<mailto:markusklyver at hotmail.com>> wrote:
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?
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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
Please excuse this indulgence.
Being on holiday I was not in a position to reply earlier.
Cheers,
Ross
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