{"id":41,"date":"2004-10-24T15:15:27","date_gmt":"2004-10-24T15:15:27","guid":{"rendered":"http:\/\/www.sixthform.info\/maths\/?p=41"},"modified":"2004-10-24T15:15:27","modified_gmt":"2004-10-24T15:15:27","slug":"proof-and-logic","status":"publish","type":"post","link":"https:\/\/www.sixthform.info\/maths\/?p=41","title":{"rendered":"Proof and logic"},"content":{"rendered":"<p>Mathematical proof is one of the topics that students find very difficult. Many of them assume what they are trying to prove, end up with a true statement and then think they have proved the result. Studying truth tables, particularly the implication operator may well help. See <a href=\"http:\/\/www.rwc.uc.edu\/koehler\/comath\/21.html\" target=\"_blank\">Logical Operations and Truth Tables<\/a><\/p>\n<p>On a mathematics forum a student wanted to use its <img src='\/maths\/latexrender\/pictures\/c51d7e23458ca0e7373a8ed6ab56b2b9.gif' title='\\LaTeX' alt='\\LaTeX' align=absmiddle> facilities (provided by <a href=\"http:\/\/www.mayer.dial.pipex.com\/tex.htm#latexrender\" target=\"_blank\">LatexRender<\/a> of course \ud83d\ude0e ) to help a friend show how to prove <\/p>\n<ul><img src='\/maths\/latexrender\/pictures\/ed46297ab00c0ebf55cdb8dea7123308.gif' title='\\sin{(x+\\frac{\\pi}{4})}+\\cos{(x+\\frac{\\pi}{4})}=\\sqrt{2}\\cos{x}' alt='\\sin{(x+\\frac{\\pi}{4})}+\\cos{(x+\\frac{\\pi}{4})}=\\sqrt{2}\\cos{x}' align=absmiddle><\/ul>\n<p>This is the original &#8216;proof&#8217; they gave. Although they have now changed it I get the impression that I failed to convince them of the faulty logic; how would you explain what is wrong?<\/p>\n<ul><img src='\/maths\/latexrender\/pictures\/ed46297ab00c0ebf55cdb8dea7123308.gif' title='\\sin{(x+\\frac{\\pi}{4})}+\\cos{(x+\\frac{\\pi}{4})}=\\sqrt{2}\\cos{x}' alt='\\sin{(x+\\frac{\\pi}{4})}+\\cos{(x+\\frac{\\pi}{4})}=\\sqrt{2}\\cos{x}' align=absmiddle><\/p>\n<p>For this problem you need to know the addition formulas:<br \/>\n<img src='\/maths\/latexrender\/pictures\/2db41e9f4735cb069331a374fe3b417c.gif' title='\\&#10;\\sin{(x+t)}=\\sin{x}\\cos{t}+\\cos{x}\\sin{t}\\&#10;\\cos{(x+t)}=\\cos{x}\\cos{t}-\\sin{x}\\sin{t}' alt='\\&#10;\\sin{(x+t)}=\\sin{x}\\cos{t}+\\cos{x}\\sin{t}\\&#10;\\cos{(x+t)}=\\cos{x}\\cos{t}-\\sin{x}\\sin{t}' align=absmiddle><\/p>\n<p>Using these formulas in the problem we can turn it into:<br \/>\n<img src='\/maths\/latexrender\/pictures\/d2eca2ceba22dcc7b75c07d740a3e3cb.gif' title='\\&#10;(\\sin{x}\\cos{\\frac{\\pi}{4}}+\\cos{x}\\sin{\\frac{\\pi}{4}})+(\\cos{x}\\cos{\\frac{\\pi}{4}}-\\sin{x}\\sin{\\frac{\\pi}{4}})=\\sqrt{2}\\cos{x}' alt='\\&#10;(\\sin{x}\\cos{\\frac{\\pi}{4}}+\\cos{x}\\sin{\\frac{\\pi}{4}})+(\\cos{x}\\cos{\\frac{\\pi}{4}}-\\sin{x}\\sin{\\frac{\\pi}{4}})=\\sqrt{2}\\cos{x}' align=absmiddle><\/p>\n<p>Then we use the fact that <img src='\/maths\/latexrender\/pictures\/7af9e6a138fed7633b2613553bb2ddc2.gif' title='\\sin{\\frac{\\pi}{4}}=\\frac{\\sqrt{2}}{2}' alt='\\sin{\\frac{\\pi}{4}}=\\frac{\\sqrt{2}}{2}' align=absmiddle> and <img src='\/maths\/latexrender\/pictures\/12a6d5e684c73d0887c40dad53f88cc5.gif' title='\\cos{\\frac{\\pi}{4}}=\\frac{\\sqrt{2}}{2}' alt='\\cos{\\frac{\\pi}{4}}=\\frac{\\sqrt{2}}{2}' align=absmiddle><\/p>\n<p>This changes it to:<br \/>\n<img src='\/maths\/latexrender\/pictures\/6a3d7dd1b635cdb0eb6378cb2a38d582.gif' title='\\&#10;\\cancel{\\frac{\\sqrt{2}}{2}\\sin{x}}}+\\frac{\\sqrt{2}}{2}\\cos{x}+\\frac{\\sqrt{2}}{2}\\cos{x}-\\cancel{\\frac{\\sqrt{2}}{2}\\sin{x}}=\\sqrt{2}\\cos{x}' alt='\\&#10;\\cancel{\\frac{\\sqrt{2}}{2}\\sin{x}}}+\\frac{\\sqrt{2}}{2}\\cos{x}+\\frac{\\sqrt{2}}{2}\\cos{x}-\\cancel{\\frac{\\sqrt{2}}{2}\\sin{x}}=\\sqrt{2}\\cos{x}' align=absmiddle><\/p>\n<p>Here we cancelled out the terms that equals zero and then adding together what we have we end up with:<\/p>\n<p><img src='\/maths\/latexrender\/pictures\/3f3237bda7e6ab7bcd5a37622cf8639d.gif' title='\\sqrt{2}\\cos{x}=\\sqrt{2}\\cos{x}' alt='\\sqrt{2}\\cos{x}=\\sqrt{2}\\cos{x}' align=absmiddle><\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Mathematical proof is one of the topics that students find very difficult. Many of them assume what they are trying to prove, end up with a true statement and then think they have proved the result. Studying truth tables, particularly the implication operator may well help. See Logical Operations and Truth Tables On a mathematics [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-41","post","type-post","status-publish","format-standard","hentry","category-articles"],"_links":{"self":[{"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=\/wp\/v2\/posts\/41","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=41"}],"version-history":[{"count":0,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=\/wp\/v2\/posts\/41\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=41"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=41"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=41"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}