{"id":33731,"date":"2023-09-01T16:53:00","date_gmt":"2023-09-01T16:53:00","guid":{"rendered":"https:\/\/www.splashlearn.com\/math-vocabulary\/?page_id=33731"},"modified":"2024-03-18T07:33:35","modified_gmt":"2024-03-18T07:33:35","slug":"reflexive-relation-definition-formula-examples-faqs","status":"publish","type":"post","link":"https:\/\/www.splashlearn.com\/math-vocabulary\/reflexive-relations","title":{"rendered":"Reflexive Relation: Definition, Formula, Examples, FAQs"},"content":{"rendered":"<div class=\"aioseo-breadcrumbs\"><span class=\"aioseo-breadcrumb\">\n\tMath-Vocabulary\n<\/span><\/div>\n\n<div class=\"ub_table-of-contents\" data-showtext=\"show\" data-hidetext=\"hide\" data-scrolltype=\"auto\" id=\"ub_table-of-contents-daed998b-d452-4307-af7f-257b650ceb9c\" data-initiallyhideonmobile=\"false\"\n                    data-initiallyshow=\"true\"><div class=\"ub_table-of-contents-extra-container\"><div class=\"ub_table-of-contents-container ub_table-of-contents-1-column \"><ul><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/reflexive-relations#0-what-is-a-reflexive-relation>What Is a Reflexive Relation?<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/reflexive-relations#2-reflexive-relation-formula>Reflexive Relation Formula<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/reflexive-relations#3-how-to-prove-a-relation-is-reflexive>How to Prove a Relation Is Reflexive<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/reflexive-relations#8-solved-examples-on-reflexive-relation>Solved Examples on Reflexive Relation<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/reflexive-relations#9-practice-problems-on-reflexive-relation>Practice Problems on Reflexive Relation<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/reflexive-relations#10-frequently-asked-questions-about-reflexive-relations>Frequently Asked Questions about Reflexive Relations<\/a><\/li><\/ul><\/div><\/div><\/div>\n\n\n<h2 class=\"wp-block-heading\" id=\"0-what-is-a-reflexive-relation\">What Is a Reflexive Relation?<\/h2>\n\n\n\n<p><strong>A Relation R defined on Set A is said to be reflexive if each element of the set is mapped to itself. In other words, aRa for all a <\/strong><strong>R.<\/strong><\/p>\n\n\n\n<p><strong>Reflexive Relation Example: <\/strong>The relation R = {(0, 0), (1, 2), (1, 1), (2, 2)} defined on A = {0, 1, 2} is a reflexive relation.\u00a0<\/p>\n\n\n\n<div id=\"recommended-games-container-id\" class=\"recommended-games-container\"><h4 class=\"recommended-games-container-headline\">Recommended Games<\/h4><div class=\"recommended-games-container-slides\"><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/relationship-between-two-patterns\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/algebra_analyze_relate_patterns_pt.png\" alt=\"Relationship Between Two Patterns Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">Relationship Between Two Patterns Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><\/div><p class=\"recommended-games-container-desc\"><a href=\"https:\/\/www.splashlearn.com\/games\">More Games<\/a><\/p><button class=\"scroll-right-arrow\"><\/button><\/div><script type=\"text\/javascript\">\n        document.addEventListener(\"DOMContentLoaded\", function() {\n            const container = document.querySelector(\"#recommended-games-container-id\");\n            const slidesContainer = container.querySelector(\".recommended-games-container-slides\");\n            const cards = slidesContainer.querySelectorAll(\".game-card-container-outer\");\n            const scrollRightArrow = container.querySelector(\".scroll-right-arrow\");\n\n            function adjustContainerStyles() {\n                const numCards = cards.length;\n\n                if (numCards === 1) {\n                    container.style.maxWidth = \"30%\";\n                    container.style.textAlign = \"center\";\n                } else if (numCards === 2) {\n                    container.style.maxWidth = \"50%\";\n                    container.style.textAlign = \"center\";\n                } else if (numCards === 3) {\n                    container.style.maxWidth = \"75%\";\n                    container.style.textAlign = \"center\";\n                } else {\n                    container.style.maxWidth = \"\";\n                    container.style.textAlign = \"\";\n                }\n            }\n\n            function checkScrollPosition() {\n                const maxScrollLeft = slidesContainer.scrollWidth - slidesContainer.clientWidth;\n                if ((slidesContainer.scrollLeft + 10) >= maxScrollLeft) {\n                    scrollRightArrow.style.display = \"none\"; \/\/ Hide the arrow if fully scrolled\n                } else {\n                    scrollRightArrow.style.display = \"block\"; \/\/ Show the arrow if not fully scrolled\n                }\n            }\n\n            scrollRightArrow.addEventListener(\"click\", function() {\n                const scrollAmount = 300; \/\/ Adjust based on the container's width and your needs\n                slidesContainer.scrollLeft += scrollAmount;\n                setTimeout(checkScrollPosition, 100); \/\/ Delay to allow scroll update\n            });\n\n            adjustContainerStyles();\n            checkScrollPosition();\n            slidesContainer.addEventListener(\"scroll\", checkScrollPosition);\n        });\n    <\/script><h2 class=\"wp-block-heading\" id=\"1-definition-of-reflexive-relation\">Definition of Reflexive Relation<\/h2>\n\n\n\n<p><strong>A binary Relation R defined on a Set A is said to be reflexive if for each element a \u2208 A, we have aRa.<\/strong><\/p>\n\n\n\n<p>R is reflexive if for each a \u2208 A, (a, a) \u2208 R.&nbsp;<\/p>\n\n\n\n<p>This proves that when each element of a set is related to itself, only then can a relation be said to be reflexive. R will not be a reflexive relation if even one element of the set is not related to itself.<\/p>\n\n\n\n<div id=\"recommended-worksheets-container-id\" class=\"recommended-games-container\"><h4 class=\"recommended-games-container-headline\">Recommended Worksheets<\/h4><div class=\"recommended-games-container-slides\"><div class=\"worksheet-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-worksheets\/understand-the-relationship-in-division\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"worksheets_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t    <div class=\"worksheet-card-container-inner-block\">\r\n\t        <img decoding=\"async\" class=\"worksheet-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/cms_assets\/s\/math-worksheets\/understand-the-relationship-in-division.jpeg\" alt=\"Understand the Relationship in Division Worksheet\">\r\n\t    <\/div>\r\n\t\t<div class=\"worksheet-card-container-inner\" >\r\n\t\t<\/div><\/a>\r\n<\/div><\/div><p class=\"recommended-games-container-desc\"><a href=\"https:\/\/www.splashlearn.com\/worksheets\">More Worksheets<\/a><\/p><button class=\"scroll-right-arrow\"><\/button><\/div><script type=\"text\/javascript\">\n        document.addEventListener(\"DOMContentLoaded\", function() {\n            const container = document.querySelector(\"#recommended-worksheets-container-id\");\n            const slidesContainer = container.querySelector(\".recommended-games-container-slides\");\n            const cards = slidesContainer.querySelectorAll(\".worksheet-card-container-outer\");\n            const scrollRightArrow = container.querySelector(\".scroll-right-arrow\");\n\n            function adjustContainerStyles() {\n                const numCards = cards.length;\n\n                if (numCards === 1) {\n                    container.style.maxWidth = \"30%\";\n                    container.style.textAlign = \"center\";\n                } else if (numCards === 2) {\n                    container.style.maxWidth = \"50%\";\n                    container.style.textAlign = \"center\";\n                } else if (numCards === 3) {\n                    container.style.maxWidth = \"75%\";\n                    container.style.textAlign = \"center\";\n                } else {\n                    container.style.maxWidth = \"\";\n                    container.style.textAlign = \"\";\n                }\n            }\n\n            function checkScrollPosition() {\n                const maxScrollLeft = slidesContainer.scrollWidth - slidesContainer.clientWidth;\n                if ((slidesContainer.scrollLeft + 10) >= maxScrollLeft) {\n                    scrollRightArrow.style.display = \"none\"; \/\/ Hide the arrow if fully scrolled\n                } else {\n                    scrollRightArrow.style.display = \"block\"; \/\/ Show the arrow if not fully scrolled\n                }\n            }\n\n            scrollRightArrow.addEventListener(\"click\", function() {\n                const scrollAmount = 300; \/\/ Adjust based on the container's width and your needs\n                slidesContainer.scrollLeft += scrollAmount;\n                setTimeout(checkScrollPosition, 100); \/\/ Delay to allow scroll update\n            });\n\n            adjustContainerStyles();\n            checkScrollPosition();\n            slidesContainer.addEventListener(\"scroll\", checkScrollPosition);\n        });\n    <\/script><h2 class=\"wp-block-heading\" id=\"2-reflexive-relation-formula\">Reflexive Relation Formula<\/h2>\n\n\n\n<p>The formula for <strong>reflexive relation<\/strong> states the number of reflexive relations in a given set.&nbsp;<\/p>\n\n\n\n<p>The formula is:&nbsp;<\/p>\n\n\n\n<p>N = 2<sup>n(n &#8211; 1)<\/sup><\/p>\n\n\n\n<p>where&nbsp;<\/p>\n\n\n\n<p>N = Number of reflexive relations<\/p>\n\n\n\n<p>n = the number of elements in the set&nbsp;<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"3-how-to-prove-a-relation-is-reflexive\">How to Prove a Relation Is Reflexive<\/h2>\n\n\n\n<p>To prove that a Relation R defined on Set A is reflexive, first identify the elements of Set A.&nbsp;<\/p>\n\n\n\n<p>For each element x of Set A, we must have an ordered pair (x, x) in R.<\/p>\n\n\n\n<p>If for a single element in A, R does not meet this condition, R is not reflexive.<\/p>\n\n\n\n<p><strong>Example:<\/strong> A = {1, 2, 3}<\/p>\n\n\n\n<p>R<sub>1<\/sub> = {(1,1), (1, 2), (2, 2), (2, 3)} is not reflexive but R<sub>2<\/sub> = {(1,1), (1,2), (2, 2), (3, 3), (1, 3)} is reflexive.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"4-properties-of-a-reflexive-relation\">Properties of a Reflexive Relation<\/h2>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Co-reflexive:<\/strong> A relation R defined on a set A is said to be a co-reflexive relation if (a, b) \u2208 R \u21d2 a = b for all a, b \u2208 A. A combination of a co-reflexive and a transitive relation will always result in a transitive.<\/li>\n\n\n\n<li><strong>Anti-reflexive:<\/strong> In set A, a relation R is anti-reflexive (or irreflexive) if no element of the set is related to itself. Thus, R is anti-reflexive if (a, a) $\\notin$ R for all a A.<\/li>\n\n\n\n<li><strong>Quasi-reflexive: <\/strong>For a given set A, the relation R will be quasi-reflexive if (a, b) \u2208 R implies that (a, a)\u2208 R and (b, b)\u2208 R for all the elements a and b of A.&nbsp;<\/li>\n\n\n\n<li><strong>Left Quasi-reflexive: <\/strong>For a given set A, the relation R will be left quasi-reflexive if (a, b)\u2208 R implies that (a, a) \u2208 R for all the elements a, b \u2208 A.<\/li>\n\n\n\n<li><strong>Right Quasi-reflexive: <\/strong>For a given set A, the relation R will be right quasi-reflexive if (a, b) \u2208 R implies that (b, b) \u2208 R for all the elements a, b \u2208 A.<\/li>\n\n\n\n<li>If there is a non-empty set A, any reflexive relation R cannot be anti-reflexive, anti-transitive, or asymmetric.<\/li>\n<\/ol>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"5-number-of-reflexive-relations\">Number of Reflexive Relations<\/h2>\n\n\n\n<p>Consider a relation R defined on set A, where the set A has \u2018n\u2019 number of elements. The elements of the relation R are ordered pairs of the form of (a, b), where a and b are elements of A.&nbsp;<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Relation R is a subset of A $\\times$ A.&nbsp;<br>If A has n elements, then A $\\times$ A has n<sup>2<\/sup> elements.&nbsp;<br>Here, in (a, b), the element \u201ca\u201d can be chosen in n ways. The element \u201cb\u201d can be chosen in n ways. So, there are n<sup>2<\/sup> ordered pairs possible for R.&nbsp;<\/li>\n<\/ul>\n\n\n\n<ul class=\"wp-block-list\">\n<li>For the Relation R to be a <strong>reflexive relation<\/strong>, R must have ordered pairs of the form (a, a) for each a\u2208 A. Since there are n elements in A, there are n such ordered pairs possible.&nbsp;<br>Thus, out of n<sup>2<\/sup> ordered pairs, n pairs must be present for a reflexive relation. We can do this straightforward selection in only 1 way. So, we select n ordered pairs in 1 way.&nbsp;<\/li>\n<\/ul>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The remaining ordered pairs are n<sup>2<\/sup> &#8211; n = n (n-1) ordered pairs, which may or may not be present. So, each such ordered pair has 2 choices (present or not present). There are 2 ways to select each element out of (n<sup>2<\/sup> &#8211; n) elements.<br>(n<sup>2<\/sup> &#8211; n) elements can be selected in 2 $\\times$ 2 $\\times$ 2$\\times$ \u2026. 2 = 2<sup>n(n-1)<\/sup>, since 2 is multiplied (n<sup>2<\/sup> &#8211; n) times.<\/li>\n<\/ul>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Number of reflexive relations = Number of ways to select n elements $\\times$ Number of ways to select remaining (n<sup>2<\/sup> &#8211; n) elements<br>Number of reflexive relations = 1 $\\times$ 2<sup>n(n-1)<\/sup> = 2<sup>n(n-1)<\/sup><\/li>\n<\/ul>\n\n\n\n<p>Therefore, the total number of reflexive relations on set A with n elements is given by N = 2<sup>n(n-1)<\/sup><sup>)<\/sup>.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"6-facts-on-reflexive-relation\">Facts on Reflexive Relation<\/h2>\n\n\n\n<div style=\"color:#0369a1;background-color:#e0f2fe\" class=\"wp-block-roelmagdaleno-callout-block has-text-color has-background is-layout-flex wp-container-roelmagdaleno-callout-block-is-layout-8cf370e7 wp-block-roelmagdaleno-callout-block-is-layout-flex\"><div>\n<ul class=\"wp-block-list\">\n<li>If any relation is symmetric, transitive and reflexive, it is known as an equivalence relation.<\/li>\n\n\n\n<li>Number of relations on set A having n elements = 2<sup>Number of elements in A x A<\/sup> = $2^{n^{2}}$<\/li>\n<\/ul>\n<\/div><\/div>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"7-conclusion\">Conclusion<\/h2>\n\n\n\n<p>In this article, we learned about reflexive relations, how to identify reflexive relations, and the formula for the number of reflexive relations on a set. Let\u2019s solve a few examples and practice problems based on these concepts.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"8-solved-examples-on-reflexive-relation\">Solved Examples on Reflexive Relation<\/h2>\n\n\n\n<p><strong>1. Is the relation R = {(0,0), (0,1)} defined on A = {0, 1} reflexive?&nbsp;<\/strong><\/p>\n\n\n\n<p><strong>Solution:&nbsp;<\/strong><\/p>\n\n\n\n<p>For R to be reflexive, we must have (m, m) R for all m A.<\/p>\n\n\n\n<p>0 \u2208 A and (0, 0) \u2208 R<\/p>\n\n\n\n<p>1 \u2208 A but (1, 1) \u2209 R<\/p>\n\n\n\n<p>Hence, R is not a reflexive relation.<\/p>\n\n\n\n<p><strong>2. If <\/strong><strong>A = {w, x, y, z}<\/strong><strong>, then find the number of reflexive relations on set A.<\/strong><\/p>\n\n\n\n<p><strong>Solution:&nbsp;<\/strong><\/p>\n\n\n\n<p><strong>A = {w, x, y, z}<\/strong>&nbsp;<\/p>\n\n\n\n<p>Number of elements in set A = n = 4&nbsp;<\/p>\n\n\n\n<p>Number of reflexive relations is given by the formula:&nbsp;<\/p>\n\n\n\n<p>N = 2<sup>n(n &#8211; 1)<\/sup><\/p>\n\n\n\n<p>N = 2<sup>4(4 &#8211; 1)<\/sup><\/p>\n\n\n\n<p>N = 2<sup>4 x <\/sup><sup>3<\/sup><\/p>\n\n\n\n<p>N = 2<sup>12<\/sup><\/p>\n\n\n\n<p>N = 4096<\/p>\n\n\n\n<p>Therefore, the number of reflexive relations in Set A = 4096.<\/p>\n\n\n\n<p><strong>3.<\/strong> <strong>A relation R is defined on the set N of natural numbers as iRj if i \u2265 j. Find out if R is a reflexive relation or not.&nbsp;<\/strong><\/p>\n\n\n\n<p><strong>Solution:&nbsp;<\/strong><\/p>\n\n\n\n<p>A relation R is defined on the set N as iRj if i \u2265 j.<\/p>\n\n\n\n<p>i = i, which satisfies i \u2265 i for every i \u2208 N.<\/p>\n\n\n\n<p>This implies that iRi.&nbsp;<\/p>\n\n\n\n<p>If i is any random element of N, we have (i, i) \u2208 R as all i \u2208 N.<\/p>\n\n\n\n<p>Hence, the relation R defined on the set N is reflexive.<\/p>\n\n\n\n<p><strong>4.<\/strong><strong> In a set of natural numbers, N, a relation R is defined as mRn only when 7m + 9n is divisible by 8. Find whether R is a reflexive relation or not.&nbsp;<\/strong><\/p>\n\n\n\n<p><strong>Solution:&nbsp;<\/strong><\/p>\n\n\n\n<p>For the relation R to be reflexive, we need (a, a) \u2208 R for all m \u2208 N<\/p>\n\n\n\n<p>R is defined as mRn only when 7m + 9n is divisible by 8.&nbsp;<\/p>\n\n\n\n<p>For m \u2208 N, we have<\/p>\n\n\n\n<p>7m + 9m = 16, which is divisible by 8.<\/p>\n\n\n\n<p>\u21d2 mRm.&nbsp;<\/p>\n\n\n\n<p>We know that m is a random element of set N, thus (m, m) \u2208 R for all m \u2208 N.<\/p>\n\n\n\n<p>Hence, R is a reflexive relation.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"9-practice-problems-on-reflexive-relation\">Practice Problems on Reflexive Relation<\/h2>\n\n\n\n<div class=\"spq_wrapper\"><h2 style=\"display:none;\">Reflexive Relation: Definition, Formula, Examples, FAQs<\/h2><p style=\"display:none;\">Attend this quiz & Test your knowledge.<\/p><div class=\"spq_question_wrapper\" data-answer=\"1\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">1<\/span><h3 class=\"sqp_question_text\">What will be the total number of reflexive relations on a set B having 3 elements?<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">32<\/div><div class=\"spq_answer_block\" data-value=\"1\">64<\/div><div class=\"spq_answer_block\" data-value=\"2\">128<\/div><div class=\"spq_answer_block\" data-value=\"3\">8<\/div><div class=\"sqp_question_hint\"><div class=\"sqp_question_hint__header\"><span class=\"spq_correct\">Correct<\/span><span class=\"spq_incorrect\">Incorrect<\/span><\/div><div class=\"sqp_question_hint__content\"><span>Correct answer is: 64<br\/>$N = 2^{n(n - 1)} = 2^{3(3 - 1)} = 2^{3\\times 2} = 2^{6} = 64$.<\/span><\/div><\/div><\/div><div class=\"spq_question_wrapper\" data-answer=\"2\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">2<\/span><h3 class=\"sqp_question_text\">What is the formula for finding the number of reflexive relations in a set?<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">$N = 2^{n(n + 1)}$<\/div><div class=\"spq_answer_block\" data-value=\"1\">$N = 2^{n}$<\/div><div class=\"spq_answer_block\" data-value=\"2\">$N = 2^{n(n \\;-\\; 1)}$<\/div><div class=\"spq_answer_block\" data-value=\"3\">$N = 2^{n(n \\;-\\; 2)}$<\/div><div class=\"sqp_question_hint\"><div class=\"sqp_question_hint__header\"><span class=\"spq_correct\">Correct<\/span><span class=\"spq_incorrect\">Incorrect<\/span><\/div><div class=\"sqp_question_hint__content\"><span>Correct answer is: $N = 2^{n(n \\;-\\; 1)}$<br\/>The formula for the number of reflexive relations is $N = 2^{n(n\\;-\\;1)}$, where n is the number of elements in the set.<\/span><\/div><\/div><\/div><div class=\"spq_question_wrapper\" data-answer=\"2\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">3<\/span><h3 class=\"sqp_question_text\">A relation R is defined for a set of integers (Z) such that aRb if and only if 2a + 4b is divisible by 3, where a, b $\\in$ Z. Is R a reflexive relation? <\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">R is not a reflexive relation.<\/div><div class=\"spq_answer_block\" data-value=\"1\">It depends on the value of a.<\/div><div class=\"spq_answer_block\" data-value=\"2\">R is a reflexive relation.<\/div><div class=\"spq_answer_block\" data-value=\"3\">It depends on the value of both a and b.<\/div><div class=\"sqp_question_hint\"><div class=\"sqp_question_hint__header\"><span class=\"spq_correct\">Correct<\/span><span class=\"spq_incorrect\">Incorrect<\/span><\/div><div class=\"sqp_question_hint__content\"><span>Correct answer is: R is a reflexive relation.<br\/>For all a \u2208 Z, we have, 2a + 4a = 6a, which is divisible by 3.<br>\r\nHere, \u2018a\u2019 will take an integer value.<br>\r\nThus, regardless of what value \u2018a\u2019 takes, R will be a reflexive relation.<\/span><\/div><\/div><\/div><\/div>  <script type=\"application\/ld+json\">{\n        \"@context\": \"https:\/\/schema.org\/\", \n        \"@type\": \"Quiz\", \n        \"typicalAgeRange\": \"3-11\",\n        \"educationalLevel\":  \"beginner\",\n        \"assesses\" : \"Attend this quiz & Test your knowledge.\",\n        \"educationalAlignment\": [\n              {\n                \"@type\": \"AlignmentObject\",\n                \"alignmentType\": \"educationalSubject\",\n                \"targetName\": \"Math\"\n              }] ,\n        \"name\": \"Reflexive Relation: Definition, Formula, Examples, FAQs\",        \n        \"about\": {\n                \"@type\": \"Thing\",\n                \"name\": \"Reflexive Relation\"\n        },  \n        \"hasPart\": [{\n                    \"@type\": \"Question\",   \n                    \"eduQuestionType\": \"Multiple choice\",\n                    \"learningResourceType\": \"Practice problem\",\n                    \"name\": \"What will be the total number of reflexive relations on a set B having 3 elements?\",\n                    \"text\": \"What will be the total number of reflexive relations on a set B having 3 elements?\",\n                    \"comment\": {\n                      \"@type\": \"Comment\",\n                      \"text\": \"$$N = 2^{n(n - 1)} = 2^{3(3 - 1)} = 2^{3\\\\times 2} = 2^{6} = 64$$.\"\n                    },\n                    \"encodingFormat\": \"text\/html\",\n                    \"suggestedAnswer\": [ {\n                                \"@type\": \"Answer\",\n                                \"position\": 0,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"32\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"$$N = 2^{n(n - 1)} = 2^{3(3 - 1)} = 2^{3\\\\times 2} = 2^{6} = 64$$.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 2,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"128\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"$$N = 2^{n(n - 1)} = 2^{3(3 - 1)} = 2^{3\\\\times 2} = 2^{6} = 64$$.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 3,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"8\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"$$N = 2^{n(n - 1)} = 2^{3(3 - 1)} = 2^{3\\\\times 2} = 2^{6} = 64$$.\"\n                                    }\n                                }],\n                    \"acceptedAnswer\": {\n                      \"@type\": \"Answer\",\n                      \"position\": 1,\n                      \"encodingFormat\": \"text\/html\",\n                      \"text\": \"64\",\n                      \"comment\": {\n                          \"@type\": \"Comment\",\n                          \"text\": \"$$N = 2^{n(n - 1)} = 2^{3(3 - 1)} = 2^{3\\\\times 2} = 2^{6} = 64$$.\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"$$N = 2^{n(n - 1)} = 2^{3(3 - 1)} = 2^{3\\\\times 2} = 2^{6} = 64$$.\"\n                      }\n                    } \n\n                    },{\n                    \"@type\": \"Question\",   \n                    \"eduQuestionType\": \"Multiple choice\",\n                    \"learningResourceType\": \"Practice problem\",\n                    \"name\": \"What is the formula for finding the number of reflexive relations in a set?\",\n                    \"text\": \"What is the formula for finding the number of reflexive relations in a set?\",\n                    \"comment\": {\n                      \"@type\": \"Comment\",\n                      \"text\": \"The formula for the number of reflexive relations is $$N = 2^{n(n\\\\;-\\\\;1)}$$, where n is the number of elements in the set.\"\n                    },\n                    \"encodingFormat\": \"text\/html\",\n                    \"suggestedAnswer\": [ {\n                                \"@type\": \"Answer\",\n                                \"position\": 0,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"$$N = 2^{n(n + 1)}$$\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"The formula for the number of reflexive relations is $$N = 2^{n(n\\\\;-\\\\;1)}$$, where n is the number of elements in the set.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 1,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"$$N = 2^{n}$$\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"The formula for the number of reflexive relations is $$N = 2^{n(n\\\\;-\\\\;1)}$$, where n is the number of elements in the set.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 3,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"$$N = 2^{n(n \\\\;-\\\\; 2)}$$\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"The formula for the number of reflexive relations is $$N = 2^{n(n\\\\;-\\\\;1)}$$, where n is the number of elements in the set.\"\n                                    }\n                                }],\n                    \"acceptedAnswer\": {\n                      \"@type\": \"Answer\",\n                      \"position\": 2,\n                      \"encodingFormat\": \"text\/html\",\n                      \"text\": \"$$N = 2^{n(n \\\\;-\\\\; 1)}$$\",\n                      \"comment\": {\n                          \"@type\": \"Comment\",\n                          \"text\": \"The formula for the number of reflexive relations is $$N = 2^{n(n\\\\;-\\\\;1)}$$, where n is the number of elements in the set.\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"The formula for the number of reflexive relations is $$N = 2^{n(n\\\\;-\\\\;1)}$$, where n is the number of elements in the set.\"\n                      }\n                    } \n\n                    },{\n                    \"@type\": \"Question\",   \n                    \"eduQuestionType\": \"Multiple choice\",\n                    \"learningResourceType\": \"Practice problem\",\n                    \"name\": \"A relation R is defined for a set of integers (Z) such that aRb if and only if 2a + 4b is divisible by 3, where a, b $$\\\\in$$ Z. Is R a reflexive relation? \",\n                    \"text\": \"A relation R is defined for a set of integers (Z) such that aRb if and only if 2a + 4b is divisible by 3, where a, b $$\\\\in$$ Z. Is R a reflexive relation? \",\n                    \"comment\": {\n                      \"@type\": \"Comment\",\n                      \"text\": \"For all a \u2208 Z, we have, 2a + 4a = 6a, which is divisible by 3.<br>\r\nHere, \u2018a\u2019 will take an integer value.<br>\r\nThus, regardless of what value \u2018a\u2019 takes, R will be a reflexive relation.\"\n                    },\n                    \"encodingFormat\": \"text\/html\",\n                    \"suggestedAnswer\": [ {\n                                \"@type\": \"Answer\",\n                                \"position\": 0,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"R is not a reflexive relation.\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"For all a \u2208 Z, we have, 2a + 4a = 6a, which is divisible by 3.<br>\r\nHere, \u2018a\u2019 will take an integer value.<br>\r\nThus, regardless of what value \u2018a\u2019 takes, R will be a reflexive relation.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 1,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"It depends on the value of a.\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"For all a \u2208 Z, we have, 2a + 4a = 6a, which is divisible by 3.<br>\r\nHere, \u2018a\u2019 will take an integer value.<br>\r\nThus, regardless of what value \u2018a\u2019 takes, R will be a reflexive relation.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 3,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"It depends on the value of both a and b.\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"For all a \u2208 Z, we have, 2a + 4a = 6a, which is divisible by 3.<br>\r\nHere, \u2018a\u2019 will take an integer value.<br>\r\nThus, regardless of what value \u2018a\u2019 takes, R will be a reflexive relation.\"\n                                    }\n                                }],\n                    \"acceptedAnswer\": {\n                      \"@type\": \"Answer\",\n                      \"position\": 2,\n                      \"encodingFormat\": \"text\/html\",\n                      \"text\": \"R is a reflexive relation.\",\n                      \"comment\": {\n                          \"@type\": \"Comment\",\n                          \"text\": \"For all a \u2208 Z, we have, 2a + 4a = 6a, which is divisible by 3.<br>\r\nHere, \u2018a\u2019 will take an integer value.<br>\r\nThus, regardless of what value \u2018a\u2019 takes, R will be a reflexive relation.\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"For all a \u2208 Z, we have, 2a + 4a = 6a, which is divisible by 3.<br>\r\nHere, \u2018a\u2019 will take an integer value.<br>\r\nThus, regardless of what value \u2018a\u2019 takes, R will be a reflexive relation.\"\n                      }\n                    } \n\n                    }]}<\/script>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"10-frequently-asked-questions-about-reflexive-relations\">Frequently Asked Questions about Reflexive Relations<\/h2>\n\n\n<div class=\"wp-block-ub-content-toggle\" id=\"ub-content-toggle-dd675637-74f2-46e5-b7c5-ab081435458c\" data-mobilecollapse=\"true\" data-desktopcollapse=\"true\">\n<div class=\"wp-block-ub-content-toggle-accordion\">\n                <div class=\"wp-block-ub-content-toggle-accordion-title-wrap\" aria-expanded=\"false\" aria-controls=\"ub-content-toggle-panel-0-dd675637-74f2-46e5-b7c5-ab081435458c\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-dd675637-74f2-46e5-b7c5-ab081435458c\"><strong>What can be the smallest reflexive relation formed on a set X = {a, b, c, d}?<\/strong><\/p><div class=\"wp-block-ub-content-toggle-accordion-toggle-wrap right\"><span class=\"wp-block-ub-content-toggle-accordion-state-indicator wp-block-ub-chevron-down\"><\/span>\n                    <\/div><\/div><div role=\"region\" class=\"wp-block-ub-content-toggle-accordion-content-wrap ub-hide\" id=\"ub-content-toggle-panel-0-dd675637-74f2-46e5-b7c5-ab081435458c\">\n\n<p>The smallest reflexive relation formed of X = {a, b, c, d} will be {(a, a), (b, b), (c, c), (d, d)}.<\/p>\n\n<\/div><\/div>\n\n<div class=\"wp-block-ub-content-toggle-accordion\">\n                <div class=\"wp-block-ub-content-toggle-accordion-title-wrap\" aria-expanded=\"false\" aria-controls=\"ub-content-toggle-panel-1-dd675637-74f2-46e5-b7c5-ab081435458c\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-dd675637-74f2-46e5-b7c5-ab081435458c\"><strong>Is every identity relation reflexive?<\/strong><\/p><div class=\"wp-block-ub-content-toggle-accordion-toggle-wrap right\"><span class=\"wp-block-ub-content-toggle-accordion-state-indicator wp-block-ub-chevron-down\"><\/span>\n                    <\/div><\/div><div role=\"region\" class=\"wp-block-ub-content-toggle-accordion-content-wrap ub-hide\" id=\"ub-content-toggle-panel-1-dd675637-74f2-46e5-b7c5-ab081435458c\">\n\n<p>Yes, every identity relation is reflexive. An identity relation on a set &#8216;A&#8217; is defined by the set of ordered pairs where each element &#8216;a&#8217; is related to itself, i.e., (a, a) for all &#8216;a&#8217; in &#8216;A&#8217;. This property of identity relations makes them reflexive.<\/p>\n\n<\/div><\/div>\n\n<div class=\"wp-block-ub-content-toggle-accordion\">\n                <div class=\"wp-block-ub-content-toggle-accordion-title-wrap\" aria-expanded=\"false\" aria-controls=\"ub-content-toggle-panel-2-dd675637-74f2-46e5-b7c5-ab081435458c\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-dd675637-74f2-46e5-b7c5-ab081435458c\"><strong>What is an example of a reflexive but not symmetric relation?<\/strong><\/p><div class=\"wp-block-ub-content-toggle-accordion-toggle-wrap right\"><span class=\"wp-block-ub-content-toggle-accordion-state-indicator wp-block-ub-chevron-down\"><\/span>\n                    <\/div><\/div><div role=\"region\" class=\"wp-block-ub-content-toggle-accordion-content-wrap ub-hide\" id=\"ub-content-toggle-panel-2-dd675637-74f2-46e5-b7c5-ab081435458c\">\n\n<p>The relation of \u201cLESS THAN OR EQUAL TO\u201d denoted by \u201c\u2264\u201d is an example of a reflexive relation which is not a symmetric relation. For all a, a \u2264 a. But, a \u2264 b does not mean that b \u2264 a.<\/p>\n\n<\/div><\/div>\n<\/div>","protected":false},"excerpt":{"rendered":"<p>What Is a Reflexive Relation? A Relation R defined on Set A is said to be reflexive if each element of the set is mapped to itself. In other words, aRa for all a R. Reflexive Relation Example: The relation R = {(0, 0), (1, 2), (1, 1), (2, 2)} defined on A = {0, &#8230; <a title=\"Reflexive Relation: Definition, Formula, Examples, FAQs\" class=\"read-more\" href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/reflexive-relations\" aria-label=\"More on Reflexive Relation: Definition, Formula, Examples, FAQs\">Read more<\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"ub_ctt_via":"","footnotes":""},"categories":[1],"tags":[],"class_list":["post-33731","post","type-post","status-publish","format-standard","hentry","category-all"],"featured_image_src":null,"author_info":{"display_name":"Mithun Jhawar","author_link":"https:\/\/www.splashlearn.com\/math-vocabulary\/author\/mithun-jhawarsplashlearn-com\/"},"acf":[],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/posts\/33731","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/comments?post=33731"}],"version-history":[{"count":8,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/posts\/33731\/revisions"}],"predecessor-version":[{"id":41085,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/posts\/33731\/revisions\/41085"}],"wp:attachment":[{"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/media?parent=33731"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/categories?post=33731"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/tags?post=33731"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}