{"id":9261,"date":"2018-02-09T12:47:56","date_gmt":"2018-02-09T07:17:56","guid":{"rendered":"http:\/\/mycbseguide.com\/blog\/?p=9261"},"modified":"2019-03-02T15:02:22","modified_gmt":"2019-03-02T09:32:22","slug":"relations-functions-class-12-notes-mathematics","status":"publish","type":"post","link":"https:\/\/mycbseguide.com\/blog\/relations-functions-class-12-notes-mathematics\/","title":{"rendered":"Relations and Functions class 12 Notes Mathematics"},"content":{"rendered":"<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_76 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Table of Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 eztoc-toggle-hide-by-default' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/mycbseguide.com\/blog\/relations-functions-class-12-notes-mathematics\/#Relations_and_Functions_Class_12_Notes_Mathematics\" >Relations and Functions Class 12 Notes Mathematics<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/mycbseguide.com\/blog\/relations-functions-class-12-notes-mathematics\/#CBSE_Class_12_Mathematics_Chapter_1_Relations_and_Functions\" >CBSE Class 12 Mathematics Chapter 1 Relations and Functions<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/mycbseguide.com\/blog\/relations-functions-class-12-notes-mathematics\/#TYPES_OF_RELATIONS\" >TYPES OF RELATIONS:<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/mycbseguide.com\/blog\/relations-functions-class-12-notes-mathematics\/#CBSE_Class_12_Revision_Notes_and_Key_Points\" >CBSE Class 12 Revision Notes and Key Points<\/a><\/li><\/ul><\/nav><\/div>\n<p><strong>Relations and Functions class 12 Notes Mathematics<\/strong> in PDF are available for free download in myCBSEguide mobile app. The best app for CBSE students now provides Relations and Functions class 12 Notes latest chapter wise notes for quick preparation of CBSE board exams and school-based annual examinations. Class 12 Mathematics notes on chapter 1 Relations and Functions are also available for download in CBSE Guide website.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Relations_and_Functions_Class_12_Notes_Mathematics\"><\/span>Relations and Functions Class 12 Notes Mathematics<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>Download CBSE class 12th revision notes for chapter 1 Relations and Functions in PDF format for free. Download revision notes for Relations and Functions class 12 Notes and score high in exams. These are the Relations and Functions class 12 Notes prepared by team of expert teachers. The revision notes help you revise the whole chapter 1 in minutes. Revision notes in exam days is one of the best tips recommended by teachers during exam days.<\/p>\n<p style=\"text-align: center;\"><strong><a class=\"button\" href=\"https:\/\/mycbseguide.com\/downloads\/cbse-class-12-mathematics\/1284\/cbse-revision-notes\/7\/\">Download Revision Notes as PDF<\/a><\/strong><\/p>\n<h2><span class=\"ez-toc-section\" id=\"CBSE_Class_12_Mathematics_Chapter_1_Relations_and_Functions\"><\/span><strong>CBSE Class 12 Mathematics Chapter 1 Relations and Functions<br \/>\n<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h3><span class=\"ez-toc-section\" id=\"TYPES_OF_RELATIONS\"><\/span><strong>TYPES OF RELATIONS:<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<ul>\n<li><strong>Empty Relation<\/strong>: It is the relation R in X given by R = <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?\\phi \\subset\" \/><\/span><\/span>\u00a0<span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?{\\rm{X}} \\times {\\rm{X}}\" \/><\/span><\/span>.<\/li>\n<li><strong>Universal Relation<\/strong>: It is the relation R in X given by R = <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?{\\rm{X}} \\times {\\rm{X}}\" \/><\/span><\/span>.<\/li>\n<li><strong>Reflexive Relation<\/strong>: A relation R in a set A is called reflexive if (a, a) \u2208 R for every a \u2208 A.<\/li>\n<li><strong>Symmetric Relation<\/strong>: A relation R in a set A is called symmetric if (<img decoding=\"async\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/revise\/12\/mathematics\/ch01\/image001.png\" \/>, <img decoding=\"async\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/revise\/12\/mathematics\/ch01\/image002.png\" \/>) \u2208 R implies that (<img decoding=\"async\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/revise\/12\/mathematics\/ch01\/image002.png\" \/>, <img decoding=\"async\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/revise\/12\/mathematics\/ch01\/image001.png\" \/>) \u2208 R, for all <img decoding=\"async\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/revise\/12\/mathematics\/ch01\/image001.png\" \/>, <img decoding=\"async\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/revise\/12\/mathematics\/ch01\/image002.png\" \/>\u00a0\u2208<\/li>\n<li><strong>Transitive Relation<\/strong>: A relation R in a set A is called transitive if (<img decoding=\"async\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/revise\/12\/mathematics\/ch01\/image001.png\" \/>, <img decoding=\"async\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/revise\/12\/mathematics\/ch01\/image002.png\" \/>) \u2208 R, and (<img decoding=\"async\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/revise\/12\/mathematics\/ch01\/image002.png\" \/>, <img decoding=\"async\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/revise\/12\/mathematics\/ch01\/image003.png\" \/>) \u2208 R together imply that all <img decoding=\"async\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/revise\/12\/mathematics\/ch01\/image001.png\" \/>, <img decoding=\"async\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/revise\/12\/mathematics\/ch01\/image002.png\" \/>, <img decoding=\"async\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/revise\/12\/mathematics\/ch01\/image003.png\" \/>\u00a0\u2208 A.<\/li>\n<li><strong>EQUIVALENCE RELATION: <\/strong>A relation R in a set A is said to be an equivalence relation if R is reflexive, symmetric and transitive.<\/li>\n<li><strong>Equivalence Classes: <\/strong>Every arbitrary equivalence relation R in a set X divides X into mutually disjoint subsets (Ai) called partitions or subdivisions of X satisfying the following conditions:<\/li>\n<\/ul>\n<p>All elements of Ai are related to each other for all i.<br \/>\nNo element of Ai is related to any element of Aj whenever i \u2260 j <img decoding=\"async\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/revise\/12\/mathematics\/ch01\/image004.png\" \/><\/p>\n<p>\u00b7\u00a0\u00a0 . These subsets (<img decoding=\"async\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/revise\/12\/mathematics\/ch01\/image005.png\" \/>) are called equivalence classes.<\/p>\n<p>\u00b7\u00a0\u00a0 For an equivalence relation in a set X, the equivalence class containing a \u2208 X, denoted by [a], is the subset of X containing all elements b related to a.<\/p>\n<p><strong>**Function: A relation f: A<\/strong> <img decoding=\"async\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/revise\/12\/mathematics\/ch01\/image006.png\" \/>B is said to be a function if every clement of A is correlated to a<\/p>\n<p>Unique element in B.<\/p>\n<p><strong>*A is domain <\/strong><\/p>\n<p><strong>* B is codomain <\/strong><\/p>\n<ul>\n<li>A <strong>function<\/strong> <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?f\" \/><\/span><\/span>\u00a0: X <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi? \\to\" \/><\/span><\/span>\u00a0Y is <strong>one-one <\/strong>(or <strong>injective<\/strong>), if <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?f\\left( {{x_1}} \\right) = f\\left( {{x_2}} \\right)\" \/><\/span><\/span>\u00a0<span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?\\Rightarrow\" \/><\/span><\/span>\u00a0<span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?{x_1} = {x_2},\" \/><\/span><\/span>\u00a0<span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?\\forall\" \/><\/span><\/span>\u00a0<span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?{x_1},{x_2} \\in {\\rm{X}}\" \/><\/span><\/span>.<\/li>\n<li>A <strong>function<\/strong> <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?f\" \/><\/span><\/span>\u00a0: X <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi? \\to\" \/><\/span><\/span>\u00a0Y is <strong>onto <\/strong>(or <strong>surjective<\/strong>), if <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?y \\in {\\rm{Y,}}\" \/><\/span><\/span>\u00a0<span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?\\exists x \\in {\\rm{X}}\" \/><\/span><\/span>\u00a0such that <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?f\\left( x \\right) = y.\" \/><\/span><\/span><\/li>\n<li>\u00a0A <strong>function<\/strong> <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?f\" \/><\/span><\/span>\u00a0: X <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi? \\to\" \/><\/span><\/span>\u00a0Y is <strong>one-one-onto <\/strong>(or <strong>bijective<\/strong>), if \u00a0is both one-one and onto.<\/li>\n<li><strong>The composition of function <\/strong><span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?f\" \/><\/span><\/span>\u00a0: A <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi? \\to\" \/><\/span><\/span>\u00a0B and <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?g\" \/><\/span><\/span>\u00a0: B <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi? \\to\" \/><\/span><\/span>\u00a0C is the function <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?gof:{\\rm{A}} \\to {\\rm{C}}\" \/><\/span><\/span>\u00a0given by <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?gof\\left( x \\right) = g\\left( {f\\left( x \\right)} \\right),\" \/><\/span><\/span>\u00a0<span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?\\forall x \\in {\\rm{A}}{\\rm{.}}\" \/><\/span><\/span><\/li>\n<li>A <strong>function<\/strong> <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?f\" \/><\/span><\/span>\u00a0: X <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi? \\to\" \/><\/span><\/span>\u00a0Y is <strong>invertible<\/strong>, if <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?\\exists g:{\\rm{Y}} \\to {\\rm{X}}\" \/><\/span><\/span>\u00a0such that\u00a0\u00a0<span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?gof = {{\\rm{I}}_x}\" \/><\/span><\/span>\u00a0and <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?fog = {{\\rm{I}}_y}.\" \/><\/span><\/span><\/li>\n<li>A <strong>function<\/strong> <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?f\" \/><\/span><\/span>\u00a0: X <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi? \\to\" \/><\/span><\/span>\u00a0Y is <strong>invertible<\/strong>, if and only if <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?f\" \/><\/span><\/span>\u00a0is one-one and onto.<\/li>\n<li><strong>Given a finite set X, a function <\/strong><span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?f\" \/><\/span><\/span>\u00a0: X <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi? \\to\" \/><\/span><\/span>\u00a0X is one-one (respectively onto) if and only if <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?f\" \/><\/span><\/span>\u00a0is onto (respectively one-one). This is the characteristics property of a finite set. This is not true for infinite set.<\/li>\n<li><strong>A binary function <\/strong>* on A is a function * from A x A to A.<\/li>\n<li><strong>An element <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?e \\in {\\rm{X}}\" \/><\/span><\/span>\u00a0is the identity element\u00a0for binary operation *<\/strong> : <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?{\\rm{X}} \\times {\\rm{X}} \\to {\\rm{X}}\" \/><\/span><\/span>, if <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?a*e=a=e*a\" \/><\/span><\/span>\u00a0<span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?\\forall a \\in {\\rm{X}}{\\rm{.}}\" \/><\/span><\/span><\/li>\n<li><b>An element <\/b><span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?e \\in {\\rm{X}}\" \/><\/span><\/span>\u00a0<strong>is invertibel for binary operation *<\/strong> : <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?{\\rm{X}} \\times {\\rm{X}} \\to {\\rm{X}}\" \/><\/span><\/span>\u00a0if there exists <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?b \\in {\\rm{X}}\" \/><\/span><\/span>\u00a0such that <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?a*b*e*b*a,\" \/><\/span><\/span>\u00a0where <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?e\" \/><\/span><\/span>\u00a0is the binary identity for the binary operation *. The element <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?b\" \/><\/span><\/span>\u00a0is called the inverse of <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?a\" \/><\/span><\/span>\u00a0and is denoted by <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?{a^{ - 1}}\" \/><\/span><\/span>.<\/li>\n<li>An operation * on X is <strong>commutative<\/strong>, if <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?a*b=b*a,\" \/><\/span><\/span>\u00a0<span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?\\forall a,b\" \/><\/span><\/span>\u00a0in X.<\/li>\n<li>An operation * on X is <strong>associative<\/strong>, if <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?(a*b)*c=a*(b*c),\" alt=\"Relations and Functions Class 12 Notes Mathematics\" width=\"159\" height=\"18\" \/><\/span><\/span>\u00a0<span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?\\forall a,b,c\" \/><\/span><\/span>\u00a0in X.<\/li>\n<\/ul>\n<h2><span class=\"ez-toc-section\" id=\"CBSE_Class_12_Revision_Notes_and_Key_Points\"><\/span><strong>CBSE Class 12 Revision Notes and Key Points<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>Relations and Functions class 12 Notes Mathematics. CBSE quick revision note for class-12 Chemistry Physics Math\u2019s, Biology and other subject are very helpful to revise the whole syllabus during exam days. The revision notes covers all important formulas and concepts given in the chapter. Even if you wish to have an overview of a chapter, quick revision notes are here to do if for you. These notes will certainly save your time during stressful exam days.<\/p>\n<ul>\n<li><a href=\"http:\/\/mycbseguide.com\/downloads\/cbse-class-12-physics\/1251\/cbse-revision-notes\/7\/\">Physics<\/a><\/li>\n<li><a href=\"http:\/\/mycbseguide.com\/downloads\/cbse-class-12-chemistry\/1267\/cbse-revision-notes\/7\/\"> Chemistry<\/a><\/li>\n<li><a href=\"http:\/\/mycbseguide.com\/downloads\/cbse-class-12-mathematics\/1284\/cbse-revision-notes\/7\/\">Mathematics<\/a><\/li>\n<li><a href=\"http:\/\/mycbseguide.com\/downloads\/cbse-class-12-biology\/1298\/cbse-revision-notes\/7\/\">Biology<\/a><\/li>\n<li><a href=\"http:\/\/mycbseguide.com\/downloads\/cbse-class-12-accountancy\/1315\/cbse-revision-notes\/7\/\">Accountancy<\/a><\/li>\n<li><a href=\"http:\/\/mycbseguide.com\/downloads\/cbse-class-12-economics\/1327\/cbse-revision-notes\/7\/\">Economics<\/a><\/li>\n<li><a href=\"http:\/\/mycbseguide.com\/downloads\/cbse-class-12-business-studies\/1727\/cbse-revision-notes\/7\/\">Business Studies<\/a><\/li>\n<li><a href=\"http:\/\/mycbseguide.com\/downloads\/cbse-class-12-computer-science\/1851\/cbse-revision-notes\/7\/\">Computer Science<\/a><\/li>\n<li><a href=\"http:\/\/mycbseguide.com\/downloads\/cbse-class-12-informatics-practices\/1873\/cbse-revision-notes\/7\/\">Informatics Practices<\/a><\/li>\n<li><a href=\"http:\/\/mycbseguide.com\/downloads\/cbse-class-12-english-core\/1855\/cbse-revision-notes\/7\/\">English Core<\/a><\/li>\n<li><a href=\"http:\/\/mycbseguide.com\/downloads\/cbse-class-12-history\/1869\/cbse-revision-notes\/7\/\">History<\/a><\/li>\n<li><a href=\"http:\/\/mycbseguide.com\/downloads\/cbse-class-12-physical-education\/1877\/cbse-revision-notes\/7\/\">Physical Education<\/a><\/li>\n<\/ul>\n<p>To download Relations and Functions class 12 Notes Mathematics, sample paper for class 12 Physics, Chemistry, Biology, History, Political Science, Economics, Geography, Computer Science, Home Science, Accountancy, Business Studies and Home Science; do check myCBSEguide app or website. myCBSEguide provides sample papers with solution, test papers for chapter-wise practice, NCERT Relations and Functions, NCERT Exemplar Relations and Functions, quick revision notes for ready reference, CBSE guess papers and CBSE important question papers. Sample Paper all are made available through\u00a0<a href=\"https:\/\/play.google.com\/store\/apps\/details?id=in.techchefs.MyCBSEGuide&amp;referrer=utm_source%3Dmycbse_bottom%26utm_medium%3Dtext%26utm_campaign%3Dmycbseads\"><strong>the best app for CBSE students<\/strong><\/a>\u00a0and myCBSEguide website<\/p>\n<ul>\n<li class=\"entry-title\"><a href=\"https:\/\/mycbseguide.com\/blog\/inverse-trigonometric-functions-class-12-notes-mathematics\/\"> Trigonometric Functions class 12 Notes Mathematics<\/a><\/li>\n<li class=\"entry-title\"><a href=\"https:\/\/mycbseguide.com\/blog\/matrices-class-12-notes-mathematics\/\">Matrices class 12 Notes Mathematics<\/a><\/li>\n<li class=\"entry-title\"><a href=\"https:\/\/mycbseguide.com\/blog\/determinants-class-12-notes-mathematics\/\">Determinants class 12 Notes Mathematics<\/a><\/li>\n<li class=\"entry-title\"><a href=\"https:\/\/mycbseguide.com\/blog\/continuity-differentiability-class-12-notes-mathematics\/\">Continuity and Differentiability class 12 Notes Mathematics<\/a><\/li>\n<li class=\"entry-title\"><a href=\"https:\/\/mycbseguide.com\/blog\/application-derivatives-class-12-notes-mathematics\/\">Application of Derivatives class 12 Notes Mathematics<\/a><\/li>\n<li class=\"entry-title\"><a href=\"https:\/\/mycbseguide.com\/blog\/integrals-class-12-notes-mathematics\/\">Integrals class 12 Notes Mathematics<\/a><\/li>\n<li class=\"entry-title\"><a href=\"https:\/\/mycbseguide.com\/blog\/application-integrals-class-12-notes-mathematics\/\">Application of Integrals class 12 Notes Mathematics<\/a><\/li>\n<li class=\"entry-title\"><a href=\"https:\/\/mycbseguide.com\/blog\/differential-equations-class-12-notes-mathematics\/\">Differential Equations class 12 Notes Mathematics<\/a><\/li>\n<li class=\"entry-title\"><a href=\"https:\/\/mycbseguide.com\/blog\/vector-algebra-class-12-notes-mathematics\/\">Vector Algebra class 12 Notes Mathematics<\/a><\/li>\n<li class=\"entry-title\"><a href=\"https:\/\/mycbseguide.com\/blog\/three-dimensional-geometry-class-12-notes-mathematics\/\">Three Dimensional Geometry class 12 Notes Mathematics<\/a><\/li>\n<li class=\"entry-title\"><a href=\"https:\/\/mycbseguide.com\/blog\/linear-programming-class-12-notes-mathematics\/\">Linear Programming class 12 Notes Mathematics<\/a><\/li>\n<li class=\"entry-title\"><a href=\"https:\/\/mycbseguide.com\/blog\/probability-class-12-notes-mathematics\/\">Probability class 12 Notes Mathematics<\/a><\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Relations and Functions class 12 Notes Mathematics in PDF are available for free download in myCBSEguide mobile app. The best app for CBSE students now provides Relations and Functions class 12 Notes latest chapter wise notes for quick preparation of CBSE board exams and school-based annual examinations. Class 12 Mathematics notes on chapter 1 Relations &#8230; <a title=\"Relations and Functions class 12 Notes Mathematics\" class=\"read-more\" href=\"https:\/\/mycbseguide.com\/blog\/relations-functions-class-12-notes-mathematics\/\" aria-label=\"More on Relations and Functions class 12 Notes Mathematics\">Read more<\/a><\/p>\n","protected":false},"author":2,"featured_media":9266,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[48,456],"tags":[457,150,461,426,240,548],"class_list":["post-9261","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-cbse-class-12","category-revision-notes","tag-cbse-notes","tag-cbse-notes-and-key-points","tag-mathematics-notes","tag-quick-revision","tag-quick-revision-notes","tag-relations-and-functions"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.0 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Relations and Functions class 12 Notes Mathematics<\/title>\n<meta name=\"description\" content=\"Relations and Functions class 12 Notes Mathematics chapter 1 in PDF format for free download. 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