{"id":32754,"date":"2023-08-06T17:14:00","date_gmt":"2023-08-06T17:14:00","guid":{"rendered":"https:\/\/www.splashlearn.com\/math-vocabulary\/?page_id=32754"},"modified":"2023-08-08T05:43:59","modified_gmt":"2023-08-08T05:43:59","slug":"symmetric-relations-definition-formula-examples-facts-faqs","status":"publish","type":"post","link":"https:\/\/www.splashlearn.com\/math-vocabulary\/symmetric-relations","title":{"rendered":"Symmetric Relations: Definition, Formula, Examples, Facts, 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-64ac4d6f-4a21-41eb-89ed-ea960ae8819f\" 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\/symmetric-relations#0-symmetric-relation-definition>Symmetric Relation Definition<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/symmetric-relations#1-how-to-check-if-a-relation-is-symmetric>How to Check if a Relation Is Symmetric<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/symmetric-relations#5-asymmetric-antisymmetric-and-symmetric-relations>Asymmetric, Antisymmetric, and Symmetric Relations<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/symmetric-relations#8-solved-examples-of-symmetric-relations>Solved Examples of Symmetric Relations<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/symmetric-relations#9-practice-problems-on-symmetric-relations>Practice Problems on Symmetric Relations<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/symmetric-relations#10-frequently-asked-questions-on-symmetric-relations>Frequently Asked Questions on Symmetric Relations<\/a><\/li><\/ul><\/div><\/div><\/div>\n\n\n<p>A relation R defined on the set A is said to be symmetric, if (x, y) is an element of R, then (y, x) is also an element of R. In other words, if x is related to y, then y is also related to x.&nbsp;<\/p>\n\n\n\n<p><strong>Example: <\/strong>$R = \\left\\{(1,5), (5,1), (4,7), (7,4)\\right\\}$ defined on the set $A = \\left\\{1, 4, 5, 7\\right\\}$ is a symmetric relation.\u00a0<\/p>\n\n\n\n<p>A relation from a set A to a set B is a subset of the cartesian product A\u00d7B, and consists of ordered pairs (a,b), where a \u2208 A and b \u2208 B. If (a,b) \u2208 R, we say that a is related to b. We can write this as aRb.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"0-symmetric-relation-definition\">Symmetric Relation Definition<\/h2>\n\n\n\n<p>The relation R defined on set A is symmetric if<\/p>\n\n\n\n<p>(p, q) \u2208 R \u21d2 (q, p) \u2208 R for all p, q \u2208 A<\/p>\n\n\n\n<p>pRq \u21d2 qRp for all p, q \u2208 A.<\/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-how-to-check-if-a-relation-is-symmetric\">How to Check if a Relation Is Symmetric<\/h2>\n\n\n\n<p>To check if a given relation is symmetric or not, we need to check if each ordered pair in the given relation satisfies the given condition.<\/p>\n\n\n\n<p>(a, b) R (b, a) R<\/p>\n\n\n\n<p><strong>Example 1: Let <\/strong>$A\u00a0 = \\left\\{0, 1, 2\\right\\}$<strong> and R be a relation defined on set A such that<\/strong><\/p>\n\n\n\n<p>$R\u00a0 = \\left\\{(0, 0), (1, 1), (2, 2), (1, 2)\\right\\}$<strong>. Is R symmetric?<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"wj-table-class\"><thead><tr><th class=\"has-text-align-center\" data-align=\"center\"><strong>(a, b)<\/strong><\/th><th class=\"has-text-align-center\" data-align=\"center\"><strong>(b, a)<\/strong><\/th><th class=\"has-text-align-center\" data-align=\"center\"><strong>Is (b, a) present in R?<\/strong><\/th><\/tr><\/thead><tbody><tr><td class=\"has-text-align-center\" data-align=\"center\">(0, 0)<\/td><td class=\"has-text-align-center\" data-align=\"center\">(0, 0)<\/td><td class=\"has-text-align-center\" data-align=\"center\">Yes<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">(1, 1)<\/td><td class=\"has-text-align-center\" data-align=\"center\">(1, 1)<\/td><td class=\"has-text-align-center\" data-align=\"center\">Yes<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">(2, 2)<\/td><td class=\"has-text-align-center\" data-align=\"center\">(2, 2)<\/td><td class=\"has-text-align-center\" data-align=\"center\">Yes<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><strong>(1, 2)<\/strong><\/td><td class=\"has-text-align-center\" data-align=\"center\"><strong>(2, 1)<\/strong><\/td><td class=\"has-text-align-center\" data-align=\"center\"><strong>No<\/strong><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p>We can see that the ordered pair (2, 1) is not present in R.&nbsp;<\/p>\n\n\n\n<p>Thus, R is not symmetric.<\/p>\n\n\n\n<p><strong>Example 2: <\/strong>$A = \\left\\{x, y, z\\right\\}$<strong> and <\/strong>$R = \\left\\{(x, y), (y, z), (x, z), (x,x), (y, x), (z, y), (z, x)\\right\\}$<\/p>\n\n\n\n<p>Observe that for each ordered pair (a, b), the corresponding ordered pair (b, a) is present in R. Thus, R is symmetric.<\/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-number-of-symmetric-relations\">Number of Symmetric Relations<\/h2>\n\n\n\n<p>We can find the total number of symmetric relations on a given set with n elements.<\/p>\n\n\n\n<p>Let us consider a set A with n elements on which a relation R is defined.<\/p>\n\n\n\n<p>The total number of symmetric relations on A is given by $2^{\\frac{n(n + 1)}{2}}$.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"3-symmetric-relation-formula\">Symmetric Relation Formula<\/h2>\n\n\n\n<p>Number of symmetric relations on a set with n elements $= N = 2^{\\frac{n(n + 1)}{2}}$<\/p>\n\n\n\n<p>where n is the number of elements in the set.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"4-symmetric-relation-examples\">Symmetric Relation Examples<\/h2>\n\n\n\n<ul class=\"wp-block-list\">\n<li>A commonly known example of symmetric relation is the relationship of biological siblings. If person X is a biological sibling to person Y, then Y is also the biological sibling of X. This is called the \u201cis a biological sibling\u201d symmetric relation.<\/li>\n\n\n\n<li>The \u201cis equal to\u201d relation is a symmetric relation since m = n, implies that n = m.&nbsp;<\/li>\n\n\n\n<li>The relation \u201cis parallel to\u201d defined on the set of straight lines is a symmetric relation since if a line l is <a href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/geometry\/parallel-lines\">parallel<\/a> to m, we know that the line m is also parallel to the line l.<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"5-asymmetric-antisymmetric-and-symmetric-relations\">Asymmetric, Antisymmetric, and Symmetric Relations<\/h2>\n\n\n\n<p>Let\u2019s understand the difference between these three types of relations.<\/p>\n\n\n\n<p><strong>Symmetric Relations<\/strong><\/p>\n\n\n\n<p>A relation is symmetric if for all a, b A, (a,b) R \u21d2 (b,a) R<\/p>\n\n\n\n<p><strong>Asymmetric Relations<\/strong><\/p>\n\n\n\n<p>A relation R on a set A is said to be asymmetric if and only if for all a, b A,<\/p>\n\n\n\n<p>(a, b) R implies that (b, a) \u2209 R,&nbsp;<\/p>\n\n\n\n<p>In simple words, we can say that an asymmetric relation is the opposite of a symmetric relation.&nbsp;<\/p>\n\n\n\n<p><strong>Example:<\/strong> The relations \u201cis less than (&lt;)\u201d, \u201cis greater than (&gt;)\u201d are asymmetric relations.<\/p>\n\n\n\n<p><strong>Antisymmetric<\/strong><strong> Relations<\/strong><\/p>\n\n\n\n<p>The relation R defined on set A is an antisymmetric relation if aRb and bRa implies that a = b.&nbsp;<\/p>\n\n\n\n<p>Simply stated, if (a, b) R and a b, then (b, a) R.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"6-facts-on-symmetric-relation\">Facts on Symmetric 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>A relation can either be symmetric or antisymmetric but not the both.<\/li>\n\n\n\n<li>Even if a single ordered pair fails to meet the condition for the symmetric relation, the relation is considered not symmetric.<\/li>\n\n\n\n<li>Number of relations on set A having n elements $= 2^{|A \\times A|} = 2^{n^{2}}$<\/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 symmetric relations, how to find a symmetric relation, and the formula to find the number of symmetric relations on a set having n elements. Let\u2019s solve a few examples based on these concepts.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"8-solved-examples-of-symmetric-relations\">Solved Examples of Symmetric Relations<\/h2>\n\n\n\n<p><strong>1. If R is a relation on a set <\/strong>$A = \\left\\{1, 2, 3\\right\\}$<strong>, where R is defined as<\/strong><\/p>\n\n\n\n<p>$R = \\left\\{(1,1), (1,2), (1,3), (2,3), (3,1)\\right\\}$<strong>, then check if R is a symmetric relation or not.<\/strong><\/p>\n\n\n\n<p><strong>Solution:&nbsp;<\/strong><\/p>\n\n\n\n<p>For the relation R to be symmetric, we must have (b, a) \u2208 R for each (a, b)\u2208 R.<\/p>\n\n\n\n<p>As we can see that (1, 2)\u2208 R.\u00a0<\/p>\n\n\n\n<p>For R to be symmetric, (2, 1) should be in R, but that\u2019s not the case.<\/p>\n\n\n\n<p>Therefore, R is not a symmetric relation.<\/p>\n\n\n\n<p><strong>2. Let <\/strong>$R = \\left\\{(a, a), (e, e), (i, i), (o, o), (u, u)\\right\\}$<strong> be a relation defined on the set <\/strong>$A = \\left\\{a, e, i , o , u\\right\\}$<strong>. Examine if R is symmetric.<\/strong><\/p>\n\n\n\n<p><strong>Solution:<\/strong>&nbsp;<\/p>\n\n\n\n<p>For the relation R to be symmetric, it should satisfy the following condition:<\/p>\n\n\n\n<p>aRb implies that bRa.&nbsp;<\/p>\n\n\n\n<p>In other words, if (a, b) \u2208 R, then (b, a) \u2208 R for all a, b in A.<\/p>\n\n\n\n<p>In $R = \\left\\{(a, a), (e, e), (i, i), (o, o), (u, u)\\right\\}$, there is no ordered pair of the form (a, b). All ordered pairs are of the form (a, a). So, all ordered pairs satisfy this condition.\u00a0<\/p>\n\n\n\n<p>Hence, R is symmetric.<\/p>\n\n\n\n<p><strong>3. Let <\/strong>$A = \\left\\{p, q, r\\right\\}$<strong> and R be a relation defined on the set A as shown:<\/strong><\/p>\n\n\n\n<p>$R = \\left\\{(p, p), (q, q), (p, r), (r, p), (r, r)\\right\\}$<\/p>\n\n\n\n<p><strong>Check if R is symmetric or not.<\/strong><\/p>\n\n\n\n<p><strong>Solution:<\/strong><\/p>\n\n\n\n<p>To check if the relation R is symmetric or not, we check the condition given for each ordered pair in R.<\/p>\n\n\n\n<p>(p, q) \u2208 R \u21d2 (q, p) \u2208 R for all p, q \u2208 A<\/p>\n\n\n\n<p>Now, let us compare the above condition for every ordered pair in R.<\/p>\n\n\n\n<p>Here, it can be seen that (p, r) \u2208 R and also (r, p) \u2208 R.\u00a0<\/p>\n\n\n\n<p>Hence, R is symmetric.<\/p>\n\n\n\n<p><strong>4. Let <\/strong>$A = \\left\\{1, 2, 3\\right\\}$<strong> and <\/strong>$B = \\left\\{(1, 1), (2, 2), (3, 3), (1, 2), (2, 3)\\right\\}$<strong>.<\/strong><\/p>\n\n\n\n<p><strong>Show that B is symmetric or not.<\/strong><\/p>\n\n\n\n<p><strong>Solution:<\/strong><\/p>\n\n\n\n<p>$A = \\left\\{1, 2, 3\\right\\}$ and $B = \\left\\{(1, 1), (2, 2), (3, 3), (1, 2), (2, 3)\\right\\}$.<\/p>\n\n\n\n<p>Here,<\/p>\n\n\n\n<p>(2, 3) \u2208 B but (3, 2)\u2209 B.<\/p>\n\n\n\n<p>(1, 2) \u2208 B but (2, 1) \u2209 B.<\/p>\n\n\n\n<p>So, B is not symmetric.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"9-practice-problems-on-symmetric-relations\">Practice Problems on Symmetric Relations<\/h2>\n\n\n\n<div class=\"spq_wrapper\"><h2 style=\"display:none;\">Symmetric Relations: Definition, Formula, Examples, Facts, FAQs<\/h2><p style=\"display:none;\">Attend this quiz & Test your knowledge.<\/p><div class=\"spq_question_wrapper\" data-answer=\"3\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">1<\/span><h3 class=\"sqp_question_text\">If A = {1, 2, 3}, the number of symmetric relations in A is<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">328<\/div><div class=\"spq_answer_block\" data-value=\"1\">324<\/div><div class=\"spq_answer_block\" data-value=\"2\">8<\/div><div class=\"spq_answer_block\" data-value=\"3\">64<\/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\/>$N = 2^{\\frac{n(n+1)}{2}} = 2^{\\frac{3\\times4}{2}} = 2^{6} = 64$<br>\r\nWe get the number of symmetric relations as 64.<\/span><\/div><\/div><\/div><div class=\"spq_question_wrapper\" data-answer=\"0\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">2<\/span><h3 class=\"sqp_question_text\">The relation R = {(1, 2), (2, 1)} on set A = {1, 2} is<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">symmetric<\/div><div class=\"spq_answer_block\" data-value=\"1\">asymmetric<\/div><div class=\"spq_answer_block\" data-value=\"2\">reflexive<\/div><div class=\"spq_answer_block\" data-value=\"3\">None of the above<\/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: symmetric<br\/>The relation R is symmetric since for (1, 2) \u2208 R, we have (2, 1) \u2208 R and vice-versa.<\/span><\/div><\/div><\/div><div class=\"spq_question_wrapper\" data-answer=\"0\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">3<\/span><h3 class=\"sqp_question_text\">The relation R is said to be symmetric if aRb implies that<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">aRa<\/div><div class=\"spq_answer_block\" data-value=\"1\">bRb<\/div><div class=\"spq_answer_block\" data-value=\"2\">bRa<\/div><div class=\"spq_answer_block\" data-value=\"3\">a = 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: aRa<br\/>The relation R is said to be symmetric if aRb implies that bRa.<\/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\": \"Symmetric Relations: Definition, Formula, Examples, Facts, FAQs\",        \n        \"about\": {\n                \"@type\": \"Thing\",\n                \"name\": \"Symmetric Relations\"\n        },  \n        \"hasPart\": [{\n                    \"@type\": \"Question\",   \n                    \"eduQuestionType\": \"Multiple choice\",\n                    \"learningResourceType\": \"Practice problem\",\n                    \"name\": \"If A = {1, 2, 3}, the number of symmetric relations in A is\",\n                    \"text\": \"If A = {1, 2, 3}, the number of symmetric relations in A is\",\n                    \"comment\": {\n                      \"@type\": \"Comment\",\n                      \"text\": \"$$N = 2^{\\\\frac{n(n+1)}{2}} = 2^{\\\\frac{3\\\\times4}{2}} = 2^{6} = 64$$<br>\r\nWe get the number of symmetric relations as 64.\"\n                    },\n                    \"encodingFormat\": \"text\/html\",\n                    \"suggestedAnswer\": [ {\n                                \"@type\": \"Answer\",\n                                \"position\": 0,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"328\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"$$N = 2^{\\\\frac{n(n+1)}{2}} = 2^{\\\\frac{3\\\\times4}{2}} = 2^{6} = 64$$<br>\r\nWe get the number of symmetric relations as 64.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 1,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"324\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"$$N = 2^{\\\\frac{n(n+1)}{2}} = 2^{\\\\frac{3\\\\times4}{2}} = 2^{6} = 64$$<br>\r\nWe get the number of symmetric relations as 64.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 2,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"8\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"$$N = 2^{\\\\frac{n(n+1)}{2}} = 2^{\\\\frac{3\\\\times4}{2}} = 2^{6} = 64$$<br>\r\nWe get the number of symmetric relations as 64.\"\n                                    }\n                                }],\n                    \"acceptedAnswer\": {\n                      \"@type\": \"Answer\",\n                      \"position\": 3,\n                      \"encodingFormat\": \"text\/html\",\n                      \"text\": \"64\",\n                      \"comment\": {\n                          \"@type\": \"Comment\",\n                          \"text\": \"$$N = 2^{\\\\frac{n(n+1)}{2}} = 2^{\\\\frac{3\\\\times4}{2}} = 2^{6} = 64$$<br>\r\nWe get the number of symmetric relations as 64.\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"$$N = 2^{\\\\frac{n(n+1)}{2}} = 2^{\\\\frac{3\\\\times4}{2}} = 2^{6} = 64$$<br>\r\nWe get the number of symmetric relations as 64.\"\n                      }\n                    } \n\n                    },{\n                    \"@type\": \"Question\",   \n                    \"eduQuestionType\": \"Multiple choice\",\n                    \"learningResourceType\": \"Practice problem\",\n                    \"name\": \"The relation R = {(1, 2), (2, 1)} on set A = {1, 2} is\",\n                    \"text\": \"The relation R = {(1, 2), (2, 1)} on set A = {1, 2} is\",\n                    \"comment\": {\n                      \"@type\": \"Comment\",\n                      \"text\": \"The relation R is symmetric since for (1, 2) \u2208 R, we have (2, 1) \u2208 R and vice-versa.\"\n                    },\n                    \"encodingFormat\": \"text\/html\",\n                    \"suggestedAnswer\": [ {\n                                \"@type\": \"Answer\",\n                                \"position\": 1,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"asymmetric\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"The relation R is symmetric since for (1, 2) \u2208 R, we have (2, 1) \u2208 R and vice-versa.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 2,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"reflexive\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"The relation R is symmetric since for (1, 2) \u2208 R, we have (2, 1) \u2208 R and vice-versa.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 3,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"None of the above\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"The relation R is symmetric since for (1, 2) \u2208 R, we have (2, 1) \u2208 R and vice-versa.\"\n                                    }\n                                }],\n                    \"acceptedAnswer\": {\n                      \"@type\": \"Answer\",\n                      \"position\": 0,\n                      \"encodingFormat\": \"text\/html\",\n                      \"text\": \"symmetric\",\n                      \"comment\": {\n                          \"@type\": \"Comment\",\n                          \"text\": \"The relation R is symmetric since for (1, 2) \u2208 R, we have (2, 1) \u2208 R and vice-versa.\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"The relation R is symmetric since for (1, 2) \u2208 R, we have (2, 1) \u2208 R and vice-versa.\"\n                      }\n                    } \n\n                    },{\n                    \"@type\": \"Question\",   \n                    \"eduQuestionType\": \"Multiple choice\",\n                    \"learningResourceType\": \"Practice problem\",\n                    \"name\": \"The relation R is said to be symmetric if aRb implies that\",\n                    \"text\": \"The relation R is said to be symmetric if aRb implies that\",\n                    \"comment\": {\n                      \"@type\": \"Comment\",\n                      \"text\": \"The relation R is said to be symmetric if aRb implies that bRa.\"\n                    },\n                    \"encodingFormat\": \"text\/html\",\n                    \"suggestedAnswer\": [ {\n                                \"@type\": \"Answer\",\n                                \"position\": 1,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"bRb\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"The relation R is said to be symmetric if aRb implies that bRa.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 2,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"bRa\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"The relation R is said to be symmetric if aRb implies that bRa.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 3,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"a = b\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"The relation R is said to be symmetric if aRb implies that bRa.\"\n                                    }\n                                }],\n                    \"acceptedAnswer\": {\n                      \"@type\": \"Answer\",\n                      \"position\": 0,\n                      \"encodingFormat\": \"text\/html\",\n                      \"text\": \"aRa\",\n                      \"comment\": {\n                          \"@type\": \"Comment\",\n                          \"text\": \"The relation R is said to be symmetric if aRb implies that bRa.\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"The relation R is said to be symmetric if aRb implies that bRa.\"\n                      }\n                    } \n\n                    }]}<\/script>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"10-frequently-asked-questions-on-symmetric-relations\">Frequently Asked Questions on Symmetric Relations<\/h2>\n\n\n<div class=\"wp-block-ub-content-toggle\" id=\"ub-content-toggle-e00fb95c-4ee4-446a-8350-b18de3674d2c\" 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-e00fb95c-4ee4-446a-8350-b18de3674d2c\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-e00fb95c-4ee4-446a-8350-b18de3674d2c\"><strong>What exactly does symmetric mean?<\/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-e00fb95c-4ee4-446a-8350-b18de3674d2c\">\n\n<p>In mathematics, this refers to the relationship between two or more elements such that if one element is related to another, then the other element is likewise related to the first element in a similar manner.<\/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-e00fb95c-4ee4-446a-8350-b18de3674d2c\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-e00fb95c-4ee4-446a-8350-b18de3674d2c\"><strong>How do we describe a relation&#8217;s range?<\/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-e00fb95c-4ee4-446a-8350-b18de3674d2c\">\n\n<p>For the relation R defined from from A to B, the range is defined as the set of all second elements of the ordered pairs of R.<\/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-e00fb95c-4ee4-446a-8350-b18de3674d2c\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-e00fb95c-4ee4-446a-8350-b18de3674d2c\"><strong>What is an equivalence 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-e00fb95c-4ee4-446a-8350-b18de3674d2c\">\n\n<p>A relation is said to be an equivalence relation if it is reflexive, symmetric, and transitive.<\/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-3-e00fb95c-4ee4-446a-8350-b18de3674d2c\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-e00fb95c-4ee4-446a-8350-b18de3674d2c\"><strong>What is the domain of a 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-3-e00fb95c-4ee4-446a-8350-b18de3674d2c\">\n\n<p>The domain of the relation R is the set of all the first elements of the ordered pairs of R.<\/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-4-e00fb95c-4ee4-446a-8350-b18de3674d2c\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-e00fb95c-4ee4-446a-8350-b18de3674d2c\"><strong>How to prove a relation is symmetric?<\/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-4-e00fb95c-4ee4-446a-8350-b18de3674d2c\">\n\n<p>To prove that a relation is symmetric, you need to show that if an element a is related to an element b, then b is also related to a. For all pairs (a, b) that belongs to R, show that if (a, b) is in R, then (b, a) must also be in R.<\/p>\n\n<\/div><\/div>\n<\/div>","protected":false},"excerpt":{"rendered":"<p>A relation R defined on the set A is said to be symmetric, if (x, y) is an element of R, then (y, x) is also an element of R. In other words, if x is related to y, then y is also related to x.&nbsp; Example: $R = \\left\\{(1,5), (5,1), (4,7), (7,4)\\right\\}$ defined on &#8230; <a title=\"Symmetric Relations: Definition, Formula, Examples, Facts, FAQs\" class=\"read-more\" href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/symmetric-relations\" aria-label=\"More on Symmetric Relations: Definition, Formula, Examples, Facts, 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-32754","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\/32754","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=32754"}],"version-history":[{"count":6,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/posts\/32754\/revisions"}],"predecessor-version":[{"id":32764,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/posts\/32754\/revisions\/32764"}],"wp:attachment":[{"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/media?parent=32754"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/categories?post=32754"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/tags?post=32754"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}