{"id":201,"date":"2020-12-05T05:05:41","date_gmt":"2020-12-05T05:05:41","guid":{"rendered":"https:\/\/www.perfectly-logical.com\/?p=201"},"modified":"2020-12-05T05:05:41","modified_gmt":"2020-12-05T05:05:41","slug":"pure-mathematics-algebra-example","status":"publish","type":"post","link":"https:\/\/www.perfectly-logical.com\/index.php\/2020\/12\/05\/pure-mathematics-algebra-example\/","title":{"rendered":"Pure Mathematics: Algebra Example"},"content":{"rendered":"\n<h3 class=\"wp-block-heading\"><span class=\"has-inline-color has-accent-color\">&#8220;When did they start putting letters in it?&#8221;<\/span><\/h3>\n\n\n\n\n\n<p>Below are the some definitions in Modern Algebra that we will use to prove the statement:&nbsp;<em>&#8220;Let \u2606 be defined on 2\u2124={2n|n\u2208\u2124} by a \u2606 b = a + b. Prove that \u27e8\u2124,\u2606\u27e9 is a group.&#8221;<\/em>&nbsp;These definitions are assumed to be true and never false. Take your time and try to fully understand what each one means.<\/p>\n\n\n\n<p><strong>Definitions<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\"><li><strong>Set: <\/strong>a well-defined collection of objects.<\/li><li><strong>Elements:<\/strong> objects of a set.<\/li><\/ul>\n\n\n\n<h4 class=\"wp-block-heading\">Brief intermission from definitions and looking more closely at sets<\/h4>\n\n\n\n<p>So, sets are just collections of objects. In our case, we are mostly interested in sets of numbers (but don&#8217;t let that stop your imagination from running wild ;-)). You have actually been working with sets of numbers your entire life. For example, we know that -1 is not a natural number but it is an integer. To express that in set notation, we would write: -1\u2209\u2115 and -1\u2208\u2124. In this case, -1 is an element of the integers but -1 is not an element of the natural numbers. Remember the old picture where the sets of numbers are contained by the set above them?<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"560\" height=\"481\" src=\"https:\/\/www.perfectly-logical.com\/wp-content\/uploads\/2020\/12\/Sets_of_Numbers-1.png\" alt=\"\" class=\"wp-image-204\" srcset=\"https:\/\/www.perfectly-logical.com\/wp-content\/uploads\/2020\/12\/Sets_of_Numbers-1.png 560w, https:\/\/www.perfectly-logical.com\/wp-content\/uploads\/2020\/12\/Sets_of_Numbers-1-300x258.png 300w\" sizes=\"auto, (max-width: 560px) 100vw, 560px\" \/><figcaption>Sets of Numbers<\/figcaption><\/figure>\n\n\n\n<p>As an example, Natural Numbers are contained in the Whole Numbers (<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.perfectly-logical.com\/wp-content\/ql-cache\/quicklatex.com-7eebe291190a2c11a38cd1164e2ce3be_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#98;&#32;&#123;&#78;&#125;&#32;&#92;&#115;&#117;&#98;&#115;&#101;&#116;&#32;&#92;&#109;&#97;&#116;&#104;&#98;&#98;&#32;&#123;&#87;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"54\" style=\"vertical-align: -1px;\"\/>) and Integers are contained in the Rational Numbers (<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.perfectly-logical.com\/wp-content\/ql-cache\/quicklatex.com-603d9a5e677ab45b15de2363e5161f59_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#98;&#32;&#123;&#90;&#125;&#32;&#92;&#115;&#117;&#98;&#115;&#101;&#116;&#32;&#92;&#109;&#97;&#116;&#104;&#98;&#98;&#32;&#123;&#81;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"49\" style=\"vertical-align: -3px;\"\/>). <em>So, each block in the picture is a set.<\/em><\/p>\n\n\n\n<p>There is also a notation when defining your own set of numbers. For example, let&#8217;s say that we want to make a set of integers that are multiples of 5. We would write {5n|n\u2208\u2124}. This means for every integer in \u2124 (which is all of them) we multiple them by 5. So, a piece of what it would look like is {&#8230;,-10,-5,0,5,10,&#8230;} where each of those numbers would be called elements. Think about how you would define a set that would look at only odd integers.<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Back to the definitions<\/h4>\n\n\n\n<ul class=\"wp-block-list\"><li><strong>Subset:<\/strong>&nbsp;A set B is a subset of a set A if every element of B is an element of A, denoted by B\u2286A or A\u2287B. If B\u2286A but B\u2260A, we will denote that by B\u2282A or A\u2283B.<\/li><li><strong>Cartesian Product:<\/strong>&nbsp;Let A and B be sets. The set A\u2a2fB={(a,b)|a\u2208A and b\u2208B} is the Cartesian Product of A and B.&nbsp;<em>(So, A and B can be any set that we can think of. Then we are pairing them up with each other to make one element of A\u2a2fB.)<\/em><\/li><li><strong>Relation:<\/strong>&nbsp;A relation between sets A and B is a subset \u211b of A\u2a2fB. For a\u2208A and b\u2208B,(a,b)\u2208\u211b is read as &#8220;a is related to b&#8221; and is written as a\u211bb.<\/li><li><strong>Function:<\/strong>&nbsp;A function \u03d5 mapping X into Y is a relation between X and Y with the property that each x\u2208X appears as the first member of exactly one ordered pair (x,y) in \u03d5. Such a function is also called a&nbsp;<strong>map<\/strong>&nbsp;or&nbsp;<strong>mapping<\/strong>&nbsp;of X into Y.<\/li><li><strong>Binary Operation:<\/strong>&nbsp;A binary operation \u2606 on a set S is a function mapping S\u2a2fS into S. For each (a,b)\u2208S\u2a2fS, we will denote the element \u2606((a,b)) of S by a\u2606b.<\/li><li><strong>Closed under the operation:<\/strong>&nbsp;Let \u2606 be a binary operation on S and let H be a subset of S. The subset H is closed under \u2606 if for all a,b\u2208H we also have a\u2606b.<\/li><\/ul>\n\n\n\n<p>All of these definitions have led us up to this point. Now, we will be looking at the meat of Group Theory in Algebra. The definition of a group&#8230;<\/p>\n\n\n\n<p><strong>Group:<\/strong> A group \u27e8G,\u2606\u27e9 is a set G, closed under a binary operation \u2606, such that the following axioms are satisfied:<\/p>\n\n\n\n<ol class=\"wp-block-list\"><li>For all a,b,c\u2208G, we have (a\u2606b)\u2606c=a\u2606(b\u2606c). (the operation is associative)<\/li><li>There is an element e in G such that for all x\u2208G, e\u2606x=x\u2606e=x. (there is an identity element)<\/li><li>Corresponding to each a\u2208G, there is an element a\u2032\u2208G such that a\u2606a\u2032=a\u2032\u2606a=e. (for each element in G there is an inverse element)<\/li><\/ol>\n\n\n\n<p>We are now armed with enough to take on our proof that was advertised above.<\/p>\n\n\n\n<h5 class=\"has-background wp-block-heading\" style=\"background-color:#adff2f\">Statement: Let \u2606 be defined on 2\u2124={2n|n\u2208\u2124} by a \u2606 b = a + b. Prove that \u27e82\u2124,\u2606\u27e9 is a group.<\/h5>\n\n\n\n<p class=\"has-background\" style=\"background-color:#adff2f\"><em>Proof:<\/em>&nbsp;Let \u2606 be defined on 2\u2124={2n|n\u2208\u2124} by a \u2606 b = a + b. Let a\u22082\u2124, b\u22082\u2124, and c\u22082\u2124. Then, a=2k, b=2p, and c=2r where k,p,r\u2208\u2124.<\/p>\n\n\n\n<p class=\"has-background\" style=\"background-color:#adff2f\">So, (a\u2606b)\u2606c=(2k+2p)+2r=2k+(2p+2r)=a\u2606(b\u2606c) since adding integers is associative. Hence, \u27e82\u2124,\u2606\u27e9 is associative.<\/p>\n\n\n\n<p class=\"has-background\" style=\"background-color:#adff2f\">Consider e=2\u20220=0. Since 0\u2208\u2124 and e=2\u20220=0, e\u22082\u2124. So, a\u2606e=2k+2\u20220=2\u20220+2k= e\u2606a=2\u20220+2k=0+2k=2k by addition of integers. Hence, a\u2606e=e\u2606a=a, making e the identity of \u27e82\u2124,\u2606\u27e9.<\/p>\n\n\n\n<p class=\"has-background\" style=\"background-color:#adff2f\">Consider a\u2032=2(-k). Since k\u2208\u2124, -1\u2208\u2124, and the multiplication of two integers is an integer, (-1)\u2022k=-k\u2208\u2124, making a\u2032\u22082\u2124. So, a\u2606a\u2032=2k+2(-k)=2k-2k=0=e. Hence, each element in 2\u2124 has an inverse element.<\/p>\n\n\n\n<p class=\"has-background\" style=\"background-color:#adff2f\">Therefore, since \u27e82\u2124,\u2606\u27e9 is associative, has an identity element, and, for each element in 2\u2124, there is an inverse element in 2\u2124. \u27e82\u2124,\u2606\u27e9 is a group.<\/p>\n\n\n\n<p>That is it! Our hard work in understanding the definitions paid off in a big way.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>&#8220;When did they start putting letters in it?&#8221; Below are the some definitions in Modern Algebra that we will use to prove the statement:&nbsp;&#8220;Let \u2606 be defined on 2\u2124={2n|n\u2208\u2124} by a \u2606 b = a + b. Prove that \u27e8\u2124,\u2606\u27e9 is a group.&#8221;&nbsp;These definitions are assumed to be true and never false. Take your time [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[6],"tags":[47,46,8,45],"class_list":["post-201","post","type-post","status-publish","format-standard","hentry","category-math","tag-algebra","tag-example","tag-math","tag-pure"],"_links":{"self":[{"href":"https:\/\/www.perfectly-logical.com\/index.php\/wp-json\/wp\/v2\/posts\/201","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.perfectly-logical.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.perfectly-logical.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.perfectly-logical.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.perfectly-logical.com\/index.php\/wp-json\/wp\/v2\/comments?post=201"}],"version-history":[{"count":9,"href":"https:\/\/www.perfectly-logical.com\/index.php\/wp-json\/wp\/v2\/posts\/201\/revisions"}],"predecessor-version":[{"id":214,"href":"https:\/\/www.perfectly-logical.com\/index.php\/wp-json\/wp\/v2\/posts\/201\/revisions\/214"}],"wp:attachment":[{"href":"https:\/\/www.perfectly-logical.com\/index.php\/wp-json\/wp\/v2\/media?parent=201"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.perfectly-logical.com\/index.php\/wp-json\/wp\/v2\/categories?post=201"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.perfectly-logical.com\/index.php\/wp-json\/wp\/v2\/tags?post=201"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}