{"id":2913,"date":"2025-06-04T15:45:57","date_gmt":"2025-06-04T15:45:57","guid":{"rendered":"https:\/\/diznr.com\/?p=2913"},"modified":"2025-06-04T15:45:57","modified_gmt":"2025-06-04T15:45:57","slug":"day-06part-03-binary-operation-caiic-axioms-closer-associative-identity-inverse-and-commutative","status":"publish","type":"post","link":"https:\/\/www.reilsolar.com\/pdf\/day-06part-03-binary-operation-caiic-axioms-closer-associative-identity-inverse-and-commutative\/","title":{"rendered":"Day 06Part 03- Binary Operation CAIIC Axioms Closer Associative Identity Inverse and Commutative."},"content":{"rendered":"<p>Day 06Part 03- Binary Operation CAIIC Axioms Closer Associative Identity Inverse and Commutative.<\/p>\n<p>[fvplayer id=&#8221;174&#8243;]<\/p>\n<p>Let&#8217;s break down <strong>Day 06 Part 03: Binary Operation \u2013 CAIIC Axioms<\/strong> step by step. The acronym <strong>CAIIC<\/strong> stands for:<\/p>\n<ul>\n<li><strong>Closure<\/strong><\/li>\n<li><strong>Associative<\/strong><\/li>\n<li><strong>Identity<\/strong><\/li>\n<li><strong>Inverse<\/strong><\/li>\n<li><strong>Commutative<\/strong><\/li>\n<\/ul>\n<p>These are <strong>axioms<\/strong> (rules) used to describe <strong>binary operations<\/strong> on a set, such as addition, multiplication, etc.<\/p>\n<hr \/>\n<h3>\ud83d\udd22 <strong>What is a Binary Operation?<\/strong><\/h3>\n<p>A <strong>binary operation<\/strong> on a set is a rule for combining any two elements of the set to form another element <strong>in the same set<\/strong>.<\/p>\n<p>Notation: If <span class=\"katex\">\u2217*<\/span> is a binary operation on a set <span class=\"katex\">SS<\/span>, then<\/p>\n<p><span class=\"katex\">\u2200a,b\u2208S,a\u2217b\u2208S\\forall a, b \\in S, \\quad a * b \\in S<\/span><\/p>\n<hr \/>\n<h3>\u2705 <strong>C \u2013 Closure Axiom<\/strong><\/h3>\n<ul>\n<li>The result of the operation on any two elements in the set must <strong>also be in the set<\/strong>.<\/li>\n<li>Example: <span class=\"katex\">Z\\mathbb{Z}<\/span> under addition \u2192 <span class=\"katex\">2+3=5\u2208Z2 + 3 = 5 \\in \\mathbb{Z}<\/span> \u2192 Closed.<\/li>\n<\/ul>\n<hr \/>\n<h3>\ud83d\udd01 <strong>A \u2013 Associative Axiom<\/strong><\/h3>\n<ul>\n<li>The grouping of elements does <strong>not<\/strong> affect the result.<\/li>\n<li><\/li>\n<\/ul>\n<p>(a * b) * c = a * (b * c)<br \/>\n]<\/p>\n<ul>\n<li>Example: <span class=\"katex\">(2+3)+4=2+(3+4)=9(2 + 3) + 4 = 2 + (3 + 4) = 9<\/span><\/li>\n<\/ul>\n<hr \/>\n<h3>\ud83c\udd94 <strong>I \u2013 Identity Axiom<\/strong><\/h3>\n<ul>\n<li>There exists an element <span class=\"katex\">e\u2208Se \\in S<\/span> such that:<\/li>\n<\/ul>\n<p><span class=\"katex\">a\u2217e=e\u2217a=a,\u2200a\u2208Sa * e = e * a = a, \\quad \\forall a \\in S<\/span><\/p>\n<ul>\n<li>Example: For addition, 0 is the identity. <span class=\"katex\">a+0=aa + 0 = a<\/span><\/li>\n<\/ul>\n<hr \/>\n<h3>\ud83d\udd01 <strong>I \u2013 Inverse Axiom<\/strong><\/h3>\n<ul>\n<li>For every element <span class=\"katex\">a\u2208Sa \\in S<\/span>, there exists an element <span class=\"katex\">b\u2208Sb \\in S<\/span> such that:<\/li>\n<\/ul>\n<p><span class=\"katex\">a\u2217b=b\u2217a=e(identity\u00a0element)a * b = b * a = e \\quad \\text{(identity element)}<\/span><\/p>\n<ul>\n<li>Example: For addition, the inverse of 5 is -5 because <span class=\"katex\">5+(\u22125)=05 + (-5) = 0<\/span><\/li>\n<\/ul>\n<hr \/>\n<h3>\ud83d\udd04 <strong>C \u2013 Commutative Axiom<\/strong><\/h3>\n<ul>\n<li>The <strong>order<\/strong> of the elements doesn\u2019t affect the result:<\/li>\n<\/ul>\n<p><span class=\"katex\">a\u2217b=b\u2217aa * b = b * a<\/span><\/p>\n<ul>\n<li>Example: <span class=\"katex\">3+5=5+3=83 + 5 = 5 + 3 = 8<\/span><\/li>\n<\/ul>\n<hr \/>\n<h3>\ud83e\udde0 Summary Table:<\/h3>\n<table style=\"width: 66.4383%;height: 144px\">\n<thead>\n<tr style=\"height: 24px\">\n<th style=\"height: 24px\">Axiom<\/th>\n<th style=\"height: 24px\">Meaning<\/th>\n<th style=\"height: 24px\">Example (Addition on <span class=\"katex\">Z\\mathbb{Z}<\/span>)<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr style=\"height: 24px\">\n<td style=\"height: 24px\">Closure<\/td>\n<td style=\"height: 24px\">Result is still in the set<\/td>\n<td style=\"height: 24px\"><span class=\"katex\">3+4=7\u2208Z3 + 4 = 7 \\in \\mathbb{Z}<\/span><\/td>\n<\/tr>\n<tr style=\"height: 24px\">\n<td style=\"height: 24px\">Associative<\/td>\n<td style=\"height: 24px\">Grouping doesn&#8217;t matter<\/td>\n<td style=\"height: 24px\"><span class=\"katex\">(1+2)+3=1+(2+3)(1+2)+3 = 1+(2+3)<\/span><\/td>\n<\/tr>\n<tr style=\"height: 24px\">\n<td style=\"height: 24px\">Identity<\/td>\n<td style=\"height: 24px\">Exists an element <span class=\"katex\">ee<\/span><\/td>\n<td style=\"height: 24px\"><span class=\"katex\">a+0=aa + 0 = a<\/span><\/td>\n<\/tr>\n<tr style=\"height: 24px\">\n<td style=\"height: 24px\">Inverse<\/td>\n<td style=\"height: 24px\">Exists an element to undo op<\/td>\n<td style=\"height: 24px\"><span class=\"katex\">a+(\u2212a)=0a + (-a) = 0<\/span><\/td>\n<\/tr>\n<tr style=\"height: 24px\">\n<td style=\"height: 24px\">Commutative<\/td>\n<td style=\"height: 24px\">Order doesn&#8217;t matter<\/td>\n<td style=\"height: 24px\"><span class=\"katex\">a+b=b+aa + b = b + a<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<hr \/>\n<p>If you&#8217;d like examples for multiplication, matrix operations, or modular arithmetic, I can include those too!<\/p>\n<h3><a href=\"https:\/\/people.tamu.edu\/~shatalov68\/\/220_Chapter_4.pdf\" target=\"_blank\" rel=\"noopener\">Day 06Part 03- Binary Operation CAIIC Axioms Closer Associative Identity Inverse and Commutative.<\/a><\/h3>\n<h3 class=\"LC20lb MBeuO DKV0Md\"><a href=\"https:\/\/gecnilokheri.ac.in\/GPContent\/Discrete%20Mathematics%20Unit4.pdf\" target=\"_blank\" rel=\"noopener\">Algebraic Structure Binary Operation on A Set OPERATIONS<\/a><\/h3>\n<h3 class=\"LC20lb MBeuO DKV0Md\"><a href=\"https:\/\/www.ijirset.com\/upload\/october\/50_Commutative.pdf\" target=\"_blank\" rel=\"noopener\">Commutative Associative Binary Operations on a Set with &#8230;<\/a><\/h3>\n<h3 class=\"LC20lb MBeuO DKV0Md\"><a href=\"https:\/\/user.math.uzh.ch\/halbeisen\/4students\/gtln\/sec1.pdf\" target=\"_blank\" rel=\"noopener\">1. The Axioms A binary operation on a set is &#8230;<\/a><\/h3>\n<h3 class=\"LC20lb MBeuO DKV0Md\"><a href=\"https:\/\/curriculumresources.edu.gh\/wp-content\/uploads\/2024\/10\/LM-additional-mathematics-section-1-TV.pdf\" target=\"_blank\" rel=\"noopener\">section 1 binary operations, sets and binomial<\/a><\/h3>\n<h3 class=\"LC20lb MBeuO DKV0Md\"><a href=\"https:\/\/thangoswbsite.yolasite.com\/resources\/BinaryOperations.pdf\" target=\"_blank\" rel=\"noopener\">Binary Operations<\/a><\/h3>\n","protected":false},"excerpt":{"rendered":"<p>Day 06Part 03- Binary Operation CAIIC Axioms Closer Associative Identity Inverse and Commutative. [fvplayer id=&#8221;174&#8243;] Let&#8217;s break down Day 06 Part 03: Binary Operation \u2013 CAIIC Axioms step by step. The acronym CAIIC stands for: Closure Associative Identity Inverse Commutative These are axioms (rules) used to describe binary operations on a set, such as addition, [&hellip;]<\/p>\n","protected":false},"author":71,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[76],"tags":[],"class_list":["post-2913","post","type-post","status-publish","format-standard","hentry","category-discrete-mathematics"],"_links":{"self":[{"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/posts\/2913","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/users\/71"}],"replies":[{"embeddable":true,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/comments?post=2913"}],"version-history":[{"count":0,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/posts\/2913\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/media?parent=2913"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/categories?post=2913"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/tags?post=2913"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}