[OS X TeX] Latex symbol for "define equal"
Martin Berggren
martin.berggren at cs.umu.se
Tue Aug 21 14:58:46 CEST 2018
Regarding the difference between a theorem and a definition: In the words of my thesis advisor: You cannot argue with a definition. (It should be said with a french accent!) A definition can be nice, practical, useful, or impractical, unnecessary, confusing, and so on, but it cannot be true or false. A theorem, however, is true.
Cheers,
> On 21 Aug 2018, at 14:48, Markus Klyver <markusklyver at hotmail.com> wrote:
>
> Depends, literature tends to use the definitions interchangeably depending on what aspect is important to subject.
>
>
> Från: MacOSX-TeX <macosx-tex-bounces at email.esm.psu.edu <mailto:macosx-tex-bounces at email.esm.psu.edu>> för Martin Berggren <martin.berggren at cs.umu.se <mailto:martin.berggren at cs.umu.se>>
> Skickat: den 21 augusti 2018 13:59
> Till: TeX on Mac OS X Mailing List
> Ämne: Re: [OS X TeX] Latex symbol for "define equal"
>
>
>> On 21 Aug 2018, at 13:47, Markus Klyver <markusklyver at hotmail.com <mailto:markusklyver at hotmail.com>> wrote:
>>
>> 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 linearindependent eigenvectors, Ax=0 only having the trivial solution, Ax=b having a solution (which is unique) for every right-hand-side b, etc.
>
> I would save that this is a theorem, not a definition. I think of a definition as a “macro”; that is, giving a short name to a mathematical property. Example: a matrix A is called positive semidefinite when x^T Ax \geq 0 for all vectors x. The point is that you in each instances when the name is used, it can be replaced by its definition.
>
Martin Berggren
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Department of Computing Science,
UMIT Research Lab
Umeå Universitet
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http://www.cs.umu.se/~martinb <http://www.cs.umu.se/~martinb>, Martin.Berggren at cs.umu.se <mailto:Martin.Berggren at cs.umu.se>
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