{"id":838680,"date":"2024-07-30T18:07:27","date_gmt":"2024-07-30T12:37:27","guid":{"rendered":"https:\/\/leverageedu.com\/discover\/?p=838680"},"modified":"2024-07-30T18:07:27","modified_gmt":"2024-07-30T12:37:27","slug":"exam-prep-relations-and-functions","status":"publish","type":"post","link":"https:\/\/leverageedu.com\/discover\/indian-exams\/exam-prep-relations-and-functions\/","title":{"rendered":"Relations and Functions:Definition,Types,Difference and Examples"},"content":{"rendered":"\n<p>\u201cRelations and Functions\u201d are the most important terms in algebra. Relations and functions are two different mathematical concepts. It is generally said, all functions are relations, but not all relations are functions. Now, this may cause you confusion. Before we go any further, let us clarify the difference between the two in this article, along with clarifying this statement.This article can act as your one stop guide in <a href=\"https:\/\/leverageedu.com\/blog\/government-exams-after-12th-science\/\"><strong>competitive exams<\/strong><\/a> such as the <a href=\"https:\/\/leverageedu.com\/discover\/indian-exams\/jee-mains-mathematics-questions\/\"><strong>JEE<\/strong><\/a>, <strong>NEET<\/strong>, <a href=\"https:\/\/leverageedu.com\/discover\/indian-exams\/exam-prep-quantitative-aptitude-syllabus-for-ssc-cgl\/\"><strong>SSC<\/strong><\/a>, <a href=\"https:\/\/leverageedu.com\/discover\/indian-exams\/management-cat-exam\/\"><strong>CAT<\/strong><\/a>, <a href=\"https:\/\/leverageedu.com\/discover\/indian-exams\/mat-exam-pattern\/\"><strong>MAT<\/strong><\/a>, PSAT, <strong>GRE<\/strong>, Olympiads, and various <a href=\"https:\/\/leverageedu.com\/discover\/indian-exams\/engineering-entrance-exams\/\"><strong>engineering entrance exams<\/strong><\/a>.<\/p>\n\n\n\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-what-is-a-function\">What is a Function?<\/h2>\n\n\n\n<p>A function is a relation that states that each input should only have one output. Alternatively, we can define a function as a particular type of relation (a collection of ordered pairs) that adheres to a rule, meaning that each X-value should only have one Y-value.<\/p>\n\n\n\n<p><strong>Example:<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-table is-style-stripes\"><table><tbody><tr><td>Domain<\/td><td>Range<\/td><\/tr><tr><td>-1<\/td><td>-3<\/td><\/tr><tr><td>1<\/td><td>3<\/td><\/tr><tr><td>3<\/td><td>9<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p><em><strong>Domain<\/strong>\u2013 <\/em>It is a set of all input (x) values, or the initial values in an ordered pair.<\/p>\n\n\n\n<p><strong><em>Range- <\/em><\/strong>It is a set of all the output (y) values in the ordered pair that are the second values.<\/p>\n\n\n\n<p>Let\u2019s take a look at it by an example:<\/p>\n\n\n\n<p>In the relation, {(-2, 4), {6, 2), (5, -5), (-2, 3)},<\/p>\n\n\n\n<p>The domain is {-2, 5, 6} and the range is {-5, 2, 3, 4}.<\/p>\n\n\n\n<p>Note: Don\u2019t repeat the numbers while writing the domain and range. Also, it should be written in increasing order.\u00a0<\/p>\n\n\n\n<p class=\"has-text-align-center has-electric-grass-gradient-background has-background has-medium-font-size\"><strong>Must Read:<a href=\"https:\/\/leverageedu.com\/discover\/indian-exams\/exam-prep-real-number-line\/\">Real Number Line: Definition and Properties<\/a><\/strong><\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-types-of-functions\">Types of Functions<\/h2>\n\n\n\n<p>The types of functions can be defined as follows in terms of relations:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong><em>An injective function or one-to-one function:<\/em><\/strong> If there is a unique element of Q for every element of P, then a function f: P \u2192 Q is considered one to one.<\/li>\n\n\n\n<li><strong><em>Many to one function:<\/em><\/strong> A function that associates two or more P elements with the same Q element.<\/li>\n\n\n\n<li><strong><em>Surjective function or Onto function: <\/em><\/strong>A function that has a pre-image in set P for each element of set Q<\/li>\n\n\n\n<li><strong><em>One-to-one correspondence or Bijective function:<\/em><\/strong> Every element of Q has a pre-image in P, and the function f matches each element of P with a discrete element of Q.<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-what-is-a-relation\">What is a Relation?<\/h2>\n\n\n\n<p>It belongs to the Cartesian product subset or just a collection of points (arranged in pairs). Stated differently, the relation between the two sets is defined as the collection of the ordered pair, where each object from the sets makes up the ordered pair.<\/p>\n\n\n\n<p>Example: {(-2, 1), (4, 3), (7, -3)}, which is typically represented in curly brackets and set notation.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-how-to-convert-a-relation-into-a-function\">How to Convert a Relation into a Function?<\/h2>\n\n\n\n<p>A function is a unique type of relation (a collection of ordered pairs) that adheres to a rule, such as every X-value should be connected to only one Y-value.<\/p>\n\n\n\n<p class=\"has-text-align-center has-electric-grass-gradient-background has-background has-medium-font-size\">Must Read:<strong><a href=\"https:\/\/leverageedu.com\/discover\/indian-exams\/exam-prep-factors-of-a-number\/\">Factors Of A Number: Definition and Properties<\/a><\/strong><\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-difference-between-relation-and-function\">Difference Between Relation and Function<\/h2>\n\n\n\n<p>Relation and function are almost same but they do differ in some way or another. To understand it better, let us see the table below:<\/p>\n\n\n\n<figure class=\"wp-block-table is-style-stripes\"><table><tbody><tr><td><strong>Relation<\/strong><\/td><td><strong>Function<\/strong><\/td><\/tr><tr><td>A relation is described as follows: Relation from non-empty set A to non-empty set B is a subset of A \u00d7 B. For example, R \u2286 A \u00d7 B<br><\/td><td>A function is a unique kind of relationship between two sets, A and B, in which every element in A is related to all or some of the elements in B, and no two elements in B are related to any element in A.<\/td><\/tr><tr><td>If the pre-image is related, it might or might not contain an image.<br><\/td><td>Every pre-image in a function needs to have an image.<\/td><\/tr><tr><td>A pre-image may contain two or more images in the case of a relation.<br><\/td><td>There cannot be two images with the same pre-image in the case of function.<\/td><\/tr><tr><td>Relation may or may not be a function.<br><\/td><td>Function will always be a relation, too.<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-how-are-relations-and-functions-related\">How are Relations and Functions Related?<\/h2>\n\n\n\n<p>It\u2019s quite sure no one has written a comprehensive proof of this statement, but the idea revolves around the fact that functions are relations in which the codomain is dependent on the domain, whereas relations do not have to be. If that sounds hazy, it\u2019s because no one has come up with a definitive explanation yet.<\/p>\n\n\n\n<p>A function has one input and one output, whereas a relationship can have multiple outputs for a single input. This is the distinction between a function and a relation. This is the fundamental element that distinguishes between function and relation.<\/p>\n\n\n\n<p>Actually, the definition of a function requires that it be a relation. Given that, all you have to do is identify one relation that is not a function. A famous formula is y = sqrt(x) when x \u2265 0. Because each value of x (other than 0) is transferred to two different values of y, the relation is not a function. However, it can be readily converted into a function by restricting the codomain to non-negative or non-positive reals.Similarly, relations like reciprocal or <a href=\"https:\/\/leverageedu.com\/discover\/indian-exams\/exam-prep-logarithm-questions\/\"><strong>logarithm<\/strong><\/a> can be converted into functions by specifying the domain or codomain to avoid exceptions.<\/p>\n\n\n\n<p>For example: Here is the diagram of a relation that is not a function.Diagram of a relation that is not a function. One reason is that 2 is the first element in more than one ordered pair. Another reason is that neither 3 nor 4 are the first element (input) of any ordered pair therein<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img  loading=\"lazy\"  decoding=\"async\"  width=\"1024\"  height=\"1024\"  src=\"data:image\/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABAQMAAAAl21bKAAAAA1BMVEUAAP+KeNJXAAAAAXRSTlMAQObYZgAAAAlwSFlzAAAOxAAADsQBlSsOGwAAAApJREFUCNdjYAAAAAIAAeIhvDMAAAAASUVORK5CYII=\"  alt=\"\"  class=\"wp-image-838695 pk-lazyload\"  style=\"width:276px;height:auto\"  data-pk-sizes=\"auto\"  data-ls-sizes=\"auto, (max-width: 1024px) 100vw, 1024px\"  data-pk-src=\"https:\/\/cdnblogdiscover.leverageedu.com\/discover\/wp-content\/uploads\/2024\/07\/Injection_keine_Injektion_1.svg_.png\"  data-pk-srcset=\"https:\/\/blogassets.leverageedu.com\/media\/uploads\/sites\/9\/2024\/07\/15075025\/Injection_keine_Injektion_1.svg_.png 1024w, https:\/\/blogassets.leverageedu.com\/media\/uploads\/sites\/9\/2024\/07\/15075025\/Injection_keine_Injektion_1.svg_-300x300.png 300w, https:\/\/blogassets.leverageedu.com\/media\/uploads\/sites\/9\/2024\/07\/15075025\/Injection_keine_Injektion_1.svg_-150x150.png 150w, https:\/\/blogassets.leverageedu.com\/media\/uploads\/sites\/9\/2024\/07\/15075025\/Injection_keine_Injektion_1.svg_-768x768.png 768w, https:\/\/blogassets.leverageedu.com\/media\/uploads\/sites\/9\/2024\/07\/15075025\/Injection_keine_Injektion_1.svg_-400x400.png 400w, https:\/\/blogassets.leverageedu.com\/media\/uploads\/sites\/9\/2024\/07\/15075025\/Injection_keine_Injektion_1.svg_-80x80.png 80w, https:\/\/blogassets.leverageedu.com\/media\/uploads\/sites\/9\/2024\/07\/15075025\/Injection_keine_Injektion_1.svg_-96x96.png 96w\" ><\/figure>\n<\/div>\n\n\n<h2 class=\"wp-block-heading\" id=\"h-relations-and-functions-examples\">Relations and Functions: Examples <\/h2>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Is A = {(2, 4), (2, 4), (3, -8), (3, -8), (3, -8)} a function?<\/li>\n<\/ol>\n\n\n\n<p>Solution: If there are any repetitions in the X-value, the relation is not a function.<\/p>\n\n\n\n<p>But,<\/p>\n\n\n\n<p>Even if the X-values are being repeated in this instance, they are still associated with the same values of Y, therefore it remains a function.<\/p>\n\n\n\n<p>Here, the point (2, 4) is written twice, and the point (3, -8) is written three times. By writing a single copy of the repeated ordered pairs, we can rewrite it. Thus, \u201cA\u201d serves a function.<\/p>\n\n\n\n<ol class=\"wp-block-list\" start=\"2\">\n<li>All functions are relations, but not all relations are functions. Justify.<\/li>\n<\/ol>\n\n\n\n<p>Solution: Let\u2019s take two sets:\u00a0<\/p>\n\n\n\n<p>A = {(-2, 1), (5, -4), (5, 3), (6, 9)}<\/p>\n\n\n\n<p>B = {(3, 9), (2, -3), (5, -2), (1, 7)}<\/p>\n\n\n\n<p>Now, in A = {(-2, 1), (5, -4), (5, 3), (6, 9)}, there is a repetition in X-values with different Y-values, and so this relation is not a function.<\/p>\n\n\n\n<p>Similarly, B = {(3, 9), (2, -3), (5, -2), (1, 7)}, in this, the value of X is different from Y, and so this relation is a function.<\/p>\n\n\n\n<p class=\"has-text-align-center has-vivid-red-color has-text-color has-link-color has-medium-font-size wp-elements-bdb8e5d19b7d1c9ba12986c4f4db0e08\"><strong>Related Posts<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-table is-style-stripes\"><table><tbody><tr><td class=\"has-text-align-center\" data-align=\"center\"><a href=\"https:\/\/leverageedu.com\/discover\/indian-exams\/exam-prep-properties-of-isosceles-triangle\/\"><strong>Properties of Isosceles Triangle<\/strong><\/a><br><\/td><td class=\"has-text-align-center\" data-align=\"center\"><a href=\"https:\/\/leverageedu.com\/discover\/indian-exams\/exam-prep-prime-numbers\/\"><strong>Basics of Prime Numbers<\/strong><\/a><\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><a href=\"https:\/\/leverageedu.com\/discover\/indian-exams\/exam-prep-how-to-find-square-root-of-a-number\/\"><strong>Square Root Of A Number<\/strong><\/a><br><\/td><td class=\"has-text-align-center\" data-align=\"center\"><a href=\"https:\/\/leverageedu.com\/discover\/indian-exams\/exam-prep-trigonometric-ratios\/\"><strong>Trigonometric Ratios <\/strong><\/a><\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><a href=\"https:\/\/leverageedu.com\/discover\/indian-exams\/exam-prep-natural-numbers\/\"><strong>Natural Numbers<\/strong><\/a><br><\/td><td class=\"has-text-align-center\" data-align=\"center\"><a href=\"https:\/\/leverageedu.com\/discover\/indian-exams\/exam-prep-commutative-property\/\"><strong>Commutative Property<\/strong><\/a><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-faqs\">FAQs<\/h2>\n\n\n\n<div class=\"schema-faq wp-block-yoast-faq-block\"><div class=\"schema-faq-section\" id=\"faq-question-1722342319535\"><strong class=\"schema-faq-question\">Is the function defined by the relation?<\/strong> <p class=\"schema-faq-answer\">If every element in set P is associated to exactly one element in set Q, then there exists a relation from set P to set Q that defines a function.<\/p> <\/div> <div class=\"schema-faq-section\" id=\"faq-question-1722342337769\"><strong class=\"schema-faq-question\">In mathematics, what are relations?<\/strong> <p class=\"schema-faq-answer\">A relation in mathematics is a set of values of ordered pairs that define a relationship between them. The items in the first set, referred to as the domain, are associated with the elements in the second set, referred to as the range.<\/p> <\/div> <div class=\"schema-faq-section\" id=\"faq-question-1722342355653\"><strong class=\"schema-faq-question\">How is a relation classified as a function?<\/strong> <p class=\"schema-faq-answer\">A function is referred to as such when each value entered into it produces a single, unique outcome. In this case, the output values are referred to as the range and the input values as the domain.<\/p> <\/div> <\/div>\n\n\n\n<p>This was all about the \u201c<strong><em>Relations and Functions<\/em>\u201d<\/strong>.\u00a0 For more such informative blogs, check out our <a href=\"https:\/\/leverageedu.com\/discover\/category\/indian-exams\/study-material\/maths\/\"><strong>Maths Section<\/strong><\/a>, or you can learn more about us by visiting our <a href=\"https:\/\/leverageedu.com\/discover\/category\/indian-exams\/study-material\/\"><strong>Study Material Section<\/strong><\/a> page.<\/p>\n","protected":false},"excerpt":{"rendered":"\u201cRelations and Functions\u201d are the most important terms in algebra. Relations and functions are two different mathematical concepts.&hellip;\n","protected":false},"author":124,"featured_media":838693,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"editor_notices":[],"footnotes":""},"categories":[369,476,396],"tags":[],"class_list":{"0":"post-838680","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"has-post-thumbnail","7":"category-indian-exams","8":"category-maths","9":"category-study-material"},"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v27.5 (Yoast SEO v27.5) - https:\/\/yoast.com\/product\/yoast-seo-premium-wordpress\/ -->\n<title>Relations and Functions:Definition,Types,Difference and Examples - Leverage Edu Discover<\/title>\n<meta name=\"description\" content=\"Read this article on Relations and Functions.Learn more about definition,types ,difference and examples related to it in detail.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/leverageedu.com\/discover\/indian-exams\/exam-prep-relations-and-functions\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Relations and Functions:Definition,Types,Difference and Examples\" \/>\n<meta property=\"og:description\" content=\"Read this article on Relations and Functions.Learn more about definition,types ,difference and examples related to it in detail.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/leverageedu.com\/discover\/indian-exams\/exam-prep-relations-and-functions\/\" \/>\n<meta property=\"og:site_name\" content=\"Leverage Edu Discover\" \/>\n<meta property=\"article:published_time\" content=\"2024-07-30T12:37:27+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/blogassets.leverageedu.com\/media\/uploads\/sites\/9\/2024\/07\/15075022\/Relations-and-Functions.jpg\" \/>\n\t<meta property=\"og:image:width\" content=\"1024\" \/>\n\t<meta property=\"og:image:height\" content=\"640\" \/>\n\t<meta property=\"og:image:type\" content=\"image\/jpeg\" \/>\n<meta name=\"author\" content=\"argima\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"argima\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"6 minutes\" \/>\n<!-- \/ Yoast SEO Premium plugin. -->","yoast_head_json":{"title":"Relations and Functions:Definition,Types,Difference and Examples - 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