<html><head><meta http-equiv="Content-Type" content="text/html; charset=utf-8"></head><body style="word-wrap: break-word; -webkit-nbsp-mode: space; line-break: after-white-space;" class=""><div class="">Regarding the difference between a theorem and a definition: In the words of my thesis advisor: <i class=""><b class="">You cannot argue with a definition</b></i>. (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.</div><div class=""><br class=""></div><div class="">Cheers,</div><br class=""><div><blockquote type="cite" class=""><div class="">On 21 Aug 2018, at 14:48, Markus Klyver <<a href="mailto:markusklyver@hotmail.com" class="">markusklyver@hotmail.com</a>> wrote:</div><br class="Apple-interchange-newline"><div class=""><div id="divtagdefaultwrapper" dir="ltr" style="caret-color: rgb(0, 0, 0); font-style: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none; font-size: 12pt; font-family: Calibri, Helvetica, sans-serif;" class=""><div style="margin-top: 0px; margin-bottom: 0px;" class="">Depends, literature tends to use the definitions interchangeably depending on what aspect is important to subject.</div><br class=""><br class=""><div style="" class=""><hr tabindex="-1" style="display: inline-block; width: 871.21875px;" class=""><div id="divRplyFwdMsg" dir="ltr" class=""><font face="Calibri, sans-serif" style="font-size: 11pt;" class=""><b class="">Från:</b><span class="Apple-converted-space"> </span>MacOSX-TeX <<a href="mailto:macosx-tex-bounces@email.esm.psu.edu" class="">macosx-tex-bounces@email.esm.psu.edu</a>> för Martin Berggren <<a href="mailto:martin.berggren@cs.umu.se" class="">martin.berggren@cs.umu.se</a>><br class=""><b class="">Skickat:</b><span class="Apple-converted-space"> </span>den 21 augusti 2018 13:59<br class=""><b class="">Till:</b><span class="Apple-converted-space"> </span>TeX on Mac OS X Mailing List<br class=""><b class="">Ämne:</b><span class="Apple-converted-space"> </span>Re: [OS X TeX] Latex symbol for "define equal"</font><div class=""> </div></div><div class="" style="word-wrap: break-word; line-break: after-white-space;"><br class=""><div class=""><blockquote type="cite" class=""><div class="">On 21 Aug 2018, at 13:47, Markus Klyver <<a href="mailto:markusklyver@hotmail.com" class="OWAAutoLink" id="LPlnk315813" previewremoved="true">markusklyver@hotmail.com</a>> wrote:</div><br class="x_Apple-interchange-newline"><div class=""><div id="x_divtagdefaultwrapper" dir="ltr" class="" style="font-style: normal; font-weight: normal; letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; text-decoration: none; font-size: 12pt; font-family: Calibri, Helvetica, sans-serif;"><div class="" style="margin-top: 0px; margin-bottom: 0px;">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"<span class="" style="font-size: 12pt;">. It turns out that this is<span class="x_Apple-converted-space"> </span></span>equivalent<span class="" style="font-size: 12pt;"> to a lot of things, among det(A) !=0, A having full rang, A having n linear</span>independent<span class="" style="font-size: 12pt;"> eigenvectors, Ax=0 only having the trivial solution, Ax=b having a solution (which is unique) for every right-hand-side b, etc.</span></div></div></div></blockquote><div class=""><br class=""></div>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<span class="Apple-converted-space"> </span><i class="">positive semidefinite</i><span class="Apple-converted-space"> </span>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. </div><div class=""><br class=""></div></div></div></div></div></blockquote></div><div class=""><div class=""><span class="Apple-style-span" style="border-collapse: separate; font-variant-ligatures: normal; font-variant-east-asian: normal; font-variant-position: normal; line-height: normal; border-spacing: 0px; -webkit-text-decorations-in-effect: none;"><span class="Apple-style-span" style="border-collapse: separate; font-variant-ligatures: normal; font-variant-east-asian: normal; font-variant-position: normal; line-height: normal; orphans: 2; text-align: -webkit-auto; widows: 2; border-spacing: 0px; -webkit-text-decorations-in-effect: none;"><div style="word-wrap: break-word; -webkit-nbsp-mode: space; line-break: after-white-space;" class=""><span class="Apple-style-span" style="border-collapse: separate; font-variant-ligatures: normal; font-variant-east-asian: normal; font-variant-position: normal; line-height: normal; text-align: -webkit-auto; border-spacing: 0px; -webkit-text-decorations-in-effect: none;"><div style="word-wrap: break-word; -webkit-nbsp-mode: space; line-break: after-white-space;" class=""><span class="Apple-style-span" style="border-collapse: separate; font-variant-ligatures: normal; font-variant-east-asian: normal; font-variant-position: normal; line-height: normal; text-align: -webkit-auto; border-spacing: 0px; -webkit-text-decorations-in-effect: none;"><div style="word-wrap: break-word; -webkit-nbsp-mode: space; line-break: after-white-space;" class=""><span class="Apple-style-span" style="border-collapse: separate; font-variant-ligatures: normal; font-variant-east-asian: normal; font-variant-position: normal; line-height: normal; text-align: -webkit-auto; border-spacing: 0px; -webkit-text-decorations-in-effect: none;"><div style="word-wrap: break-word; -webkit-nbsp-mode: space; line-break: after-white-space;" class=""><div class="">Martin Berggren</div><div class="">-------------------------------------------------------------------</div><div class="">Department of Computing Science, </div><div class="">UMIT Research Lab</div><div class="">Umeå Universitet<br class="">Campustorget 5, S-901 87 Umeå, Sweden. Ph: +46-70-732 8111<br class=""><a href="http://www.cs.umu.se/~martinb" class="">http://www.cs.umu.se/~martinb</a>, <a href="mailto:Martin.Berggren@cs.umu.se" class="">Martin.Berggren@cs.umu.se</a></div><div class=""><br class=""></div></div></span></div></span></div></span></div></span></span></div></div></body></html>