{"id":34322,"date":"2023-09-24T18:27:11","date_gmt":"2023-09-24T18:27:11","guid":{"rendered":"https:\/\/www.splashlearn.com\/math-vocabulary\/?page_id=34322"},"modified":"2024-02-05T16:17:58","modified_gmt":"2024-02-05T16:17:58","slug":"transitive-relations","status":"publish","type":"post","link":"https:\/\/www.splashlearn.com\/math-vocabulary\/transitive-relations","title":{"rendered":"Transitive Relations &#8211; Definition, Properties, 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-fce1438b-b0d8-402b-9663-7556ba50c289\" 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\/transitive-relations#0-what-are-transitive-relations>What Are Transitive Relations?<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/transitive-relations#2-examples-of-transitive-relations>Examples of Transitive Relations<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/transitive-relations#4-properties-of-transitive-relations>Properties of Transitive Relations<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/transitive-relations#7-solved-examples-on-transitive-relations>Solved Examples on Transitive Relations<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/transitive-relations#8-practice-problems-on-transitive-relations>Practice Problems on Transitive Relations<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/transitive-relations#9-frequently-asked-questions-about-transitive-relations>Frequently Asked Questions about Transitive Relations<\/a><\/li><\/ul><\/div><\/div><\/div>\n\n\n<h2 class=\"wp-block-heading\" id=\"0-what-are-transitive-relations\">What Are Transitive Relations?<\/h2>\n\n\n\n<p><strong>A relation is said to be a transitive relation, if \u201ca is related to b\u201d and \u201cb is related to c\u201d implies that \u201ca is related to c.\u201d<\/strong><\/p>\n\n\n\n<p>In other words, a relation is said to be a transitive relation, if the first element is related to the second element, and the second element is related to the third element in the set, then the first element must also be related to the third element.<\/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-transitive-relation-definition\">Transitive Relation: Definition<\/h2>\n\n\n\n<p>A binary relation R defined on <a href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/sets\">set<\/a> A is considered a transitive relation, if for the elements a, b, and c in set A, (a, b) \u2208 R, and (b, c) \u2208 R implies that (a, c) \u2208 R.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"620\" height=\"315\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/09\/transitive-relation-visual-example.png\" alt=\"Transitive relation visual example\" class=\"wp-image-34327\" title=\"Transitive relation visual example\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/09\/transitive-relation-visual-example.png 620w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/09\/transitive-relation-visual-example-300x152.png 300w\" sizes=\"auto, (max-width: 620px) 100vw, 620px\" \/><\/figure>\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-examples-of-transitive-relations\">Examples of Transitive Relations<\/h2>\n\n\n\n<p>Let\u2019s go through a few examples to understand the concept of transitive relations.<\/p>\n\n\n\n<p><strong>Example 1:<\/strong> The <a href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/counting-and-comparison\/equal-sign\">equality (=) <\/a>relation defined on the set of real numbers is a transitive relation.<\/p>\n\n\n\n<p>It is defined as&nbsp;<\/p>\n\n\n\n<p>aRb iff a = b.<\/p>\n\n\n\n<p>or&nbsp;<\/p>\n\n\n\n<p>(a, b) \u2208 R if and only if a = b.<\/p>\n\n\n\n<p>We know that if a = b, and b = c, then a = c.<\/p>\n\n\n\n<p>Thus, R is a transitive relation.<\/p>\n\n\n\n<p>This is also known as the transitive property of equality.&nbsp;<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"620\" height=\"332\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/09\/transitive-property-of-equality.png\" alt=\"Transitive property of equality\" class=\"wp-image-34328\" title=\"Transitive property of equality\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/09\/transitive-property-of-equality.png 620w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/09\/transitive-property-of-equality-300x161.png 300w\" sizes=\"auto, (max-width: 620px) 100vw, 620px\" \/><\/figure>\n\n\n\n<p><strong>Example 2:<\/strong> <strong>\u201cis subset of\u201d<\/strong> is a transitive relation defined on the power set of a set.<\/p>\n\n\n\n<p>We know that if A \u2286 B, B \u2286 C, then all the elements of A are also present in C.<\/p>\n\n\n\n<p>Thus, A \u2286 C.<\/p>\n\n\n\n<p>Thus, R is transitive.<\/p>\n\n\n\n<p><strong>Example 3: <\/strong>&#8216;<a href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/counting-and-comparison\/less-than\">Is less than<\/a>&#8216; is a transitive relation defined on a set of real numbers.&nbsp;<\/p>\n\n\n\n<p>For any real numbers a, b, and c, we have&nbsp;<\/p>\n\n\n\n<p>If a &lt; b and b &lt; c, then a &lt; c.<\/p>\n\n\n\n<p>Similarly, the relations <a href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/number-sense\/greater-than\">greater than (&gt;)<\/a>, <a href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/less-than-or-equal-to#:~:text=Less%20than%20is%20represented%20by,less%20than%20the%20given%20limit.\">less than or equal to ()<\/a>, <a href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/greater-than-or-equal-to#:~:text=The%20greater%20than%20or%20equal,less%20than%2C%20and%20minimum.%E2%80%9D\">greater than or equal to ()<\/a> are also transitive.&nbsp;<\/p>\n\n\n\n<p><strong>Example 4:<\/strong> \u201cis <a href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/geometry\/parallel-lines\">parallel<\/a> to\u201d is a transitive relation defined on the set of straight lines.<\/p>\n\n\n\n<p>If line l || line m, and line m || line n, then line l || line n.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"620\" height=\"314\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/09\/parallel-lines-l-m-and-n.png\" alt=\"Parallel lines l, m, and n\" class=\"wp-image-34329\" title=\"Parallel lines l, m, and n\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/09\/parallel-lines-l-m-and-n.png 620w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/09\/parallel-lines-l-m-and-n-300x152.png 300w\" sizes=\"auto, (max-width: 620px) 100vw, 620px\" \/><\/figure>\n\n\n\n<p><strong>Example 4: <\/strong>\u201cIs <a href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/divisibility-rules\">divisible<\/a> by: relation defined on the set of positive integers is a transitive relations.<\/p>\n\n\n\n<p>If a is divisible by b, then a = bk for some positive integer k<\/p>\n\n\n\n<p>If b is divisible by c, then b = ct for some positive integer t.<\/p>\n\n\n\n<p>We get a = c(kt) for some positive integer kt<\/p>\n\n\n\n<p>Thus, a is divisible by c.<\/p>\n\n\n\n<p>Let R = {(g, h), (h, i), (g, i)} represent a transitive relation on the set A ={g, h, i}.<\/p>\n\n\n\n<p>Here, (a, b) \u2208 R, (b, c) \u2208 R&nbsp;\u21d2 (a, c) \u2208 R&nbsp; &nbsp; for all a, b \u2208 A.<\/p>\n\n\n\n<p>g is related to h, and h is related to i.&nbsp;<\/p>\n\n\n\n<p>For R to be transitive, g must be related to i.<\/p>\n\n\n\n<p>The element (g, i) is also present in R.<\/p>\n\n\n\n<p>Thus, R is transitive.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"3-anti-transitive-relation-and-intransitive-relation\">Anti-transitive Relation and Intransitive Relation<\/h2>\n\n\n\n<p><strong>Anti-transitive Relation<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>In anti- transitive relation, if (a, b) \u2208 R and (b, c) \u2208 R, then it implies that (a, c) \u2209 R.<\/li>\n\n\n\n<li>An anti-transitive relation is a relation that violates the transitive property (a relation that does not satisfy transitive property).<\/li>\n\n\n\n<li>In an anti-transitive relation, if element a is related to element b, and element b is related to element c, then it implies that element a is not related to element c.<\/li>\n\n\n\n<li>Here, the absence of a relationship between a and c is guaranteed based on the relationship between a and b, and b and c.<\/li>\n<\/ul>\n\n\n\n<p><strong>Intransitive Relation<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>In intransitive relation, for some a, b, c in R, (a, b) \u2208 R and (b, c) \u2208 R, but (a, c) \u2209 R. Such a conclusion does not necessarily follow for each set of a, b, and c.<\/li>\n\n\n\n<li>These relations lack a consistent transitive pattern.<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"4-properties-of-transitive-relations\">Properties of Transitive Relations<\/h2>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The inverse of a transitive relation is a transitive relation.<\/li>\n\n\n\n<li>The union of two transitive relations need not be transitive.&nbsp;<\/li>\n\n\n\n<li>The intersection of two transitive relations is a transitive relation.&nbsp;<\/li>\n\n\n\n<li>A transitive relation is an asymmetric relation if and only if it is irreflexive.<\/li>\n\n\n\n<li>Transitive relations can be reflexive, meaning that every element is only related to itself. Note that it is not a requirement for transitivity.<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"5-facts-about-transitive-relations\">Facts about Transitive Relations<\/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>Whenever a relation is defined on an empty set, it is transitive.<\/li>\n\n\n\n<li>Suppose we have an ordered pair (a, b) in R, for which the ordered pair (b, c) is not present, then we don&#8217;t need to check transitivity for that ordered pair. Thus we only need to check transitivity if we discover both (a, b) and (b, c) in R.<\/li>\n<\/ul>\n<\/div><\/div>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"6-conclusion\">Conclusion<\/h2>\n\n\n\n<p>In this article, we learned about transitive relations, definition, their examples, and properties. Let\u2019s solve a few examples based on transitive relations.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"7-solved-examples-on-transitive-relations\">Solved Examples on Transitive Relations<\/h2>\n\n\n\n<p><strong>Example 1: If R is a relation on the set&nbsp; A = {2, 3, 4} given by R&nbsp; = {(2, 3), (3, 4), (2, 4)}. Verify if R is transitive.&nbsp;<\/strong><\/p>\n\n\n\n<p><strong>Solution:<\/strong><\/p>\n\n\n\n<p>Let a = 2, b = 3, c = 4.<\/p>\n\n\n\n<p>We have&nbsp;<\/p>\n\n\n\n<p>(a, b)&nbsp; =&nbsp; (2, 3) \u2208 R,&nbsp; (b, c)&nbsp; =&nbsp; (3, 4) \u2208 R<\/p>\n\n\n\n<p>(a, c)&nbsp; =&nbsp; (2, 4) \u2208 R<\/p>\n\n\n\n<p>Hence, R is transitive.<\/p>\n\n\n\n<p><strong>Example 2: If A = {1, 2, 3} and R represents a relation on the set A as R = {(1, 1), (2, 2), (3, 3), (1, 2)}. Find whether R is a transitive relation or not.&nbsp;<\/strong><\/p>\n\n\n\n<p><strong>Solution:<\/strong><\/p>\n\n\n\n<p>Relation R defined on the set A is transitive if for the elements a, b, and c in set A,&nbsp;<\/p>\n\n\n\n<p>(a, b) \u2208 R, and (b, c) \u2208 R implies that (a, c) \u2208 R.<\/p>\n\n\n\n<p>Let us check the given condition for each ordered pair.<\/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, c)<\/strong><\/th><th class=\"has-text-align-center\" data-align=\"center\"><strong>(a, c)<\/strong><\/th><\/tr><\/thead><tbody><tr><td class=\"has-text-align-center\" data-align=\"center\">(1, 1)<\/td><td class=\"has-text-align-center\" data-align=\"center\">(1, 2)<\/td><td class=\"has-text-align-center\" data-align=\"center\">(1, 2)<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">(2, 2)<\/td><td class=\"has-text-align-center\" data-align=\"center\">&#8211;<\/td><td class=\"has-text-align-center\" data-align=\"center\">&#8211;<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">(3, 3)<\/td><td class=\"has-text-align-center\" data-align=\"center\">&#8211;<\/td><td class=\"has-text-align-center\" data-align=\"center\">&#8211;<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">(1, 2)<\/td><td class=\"has-text-align-center\" data-align=\"center\">(2, 2)<\/td><td class=\"has-text-align-center\" data-align=\"center\">(1, 2)<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p>From the above table, we clearly understand that R follows the transitive rule.&nbsp;<\/p>\n\n\n\n<p><strong>Example 3: Find x if x = y and y = 5.<\/strong><\/p>\n\n\n\n<p><strong>Solution:<\/strong><\/p>\n\n\n\n<p>The \u201cis equal to\u201d relation defined on the set of real numbers is transitive.<\/p>\n\n\n\n<p>By the transitive property of equality, if a = b and b = c, then a = c.<\/p>\n\n\n\n<p>Thus, if x = y and y = 5, we have x = 5 by the transitive property.<\/p>\n\n\n\n<p><strong>Example 4:<\/strong> <strong>Is the relation \u201cis perpendicular to\u201d defined on the set of straight lines transitive?<\/strong><\/p>\n\n\n\n<p><strong>Solution:<\/strong><\/p>\n\n\n\n<p>If line m \u27c2 line p, and line p \u27c2 line n, it does not follow that line m is perpendicular to line n.&nbsp;<\/p>\n\n\n\n<p>Thus, the relation is not transitive.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"309\" height=\"161\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/09\/line-p-perpendicular-to-the-parallel-lines-m-and-n.png\" alt=\"Line p perpendicular to the parallel lines m and n\" class=\"wp-image-34330\" title=\"Line p perpendicular to the parallel lines m and n\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/09\/line-p-perpendicular-to-the-parallel-lines-m-and-n.png 309w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/09\/line-p-perpendicular-to-the-parallel-lines-m-and-n-300x156.png 300w\" sizes=\"auto, (max-width: 309px) 100vw, 309px\" \/><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"8-practice-problems-on-transitive-relations\">Practice Problems on Transitive Relations<\/h2>\n\n\n\n<div class=\"spq_wrapper\"><h2 style=\"display:none;\">Transitive Relations - Definition, Properties, Examples, FAQs<\/h2><p style=\"display:none;\">Attend this quiz & Test your knowledge.<\/p><div class=\"spq_question_wrapper\" data-answer=\"2\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">1<\/span><h3 class=\"sqp_question_text\">Which of the following relations is not a transitive relation?<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">Congruence relation on set of triangles<\/div><div class=\"spq_answer_block\" data-value=\"1\">\u201cIs equal to\u201d relation on set of real numbers<\/div><div class=\"spq_answer_block\" data-value=\"2\">\u201cIs perpendicular to\u201d relation on the set of straight lines<\/div><div class=\"spq_answer_block\" data-value=\"3\">\u201cIs less than\u201d relation on set of real numbers<\/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: \u201cIs perpendicular to\u201d relation on the set of straight lines<br\/>The Is perpendicular to relation on the set of straight lines is not transitive.<\/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\">In a transitive relation, if a is related to b, and b is related to c, then which of the following is true?<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">a is related to c<\/div><div class=\"spq_answer_block\" data-value=\"1\">c is not related to a<\/div><div class=\"spq_answer_block\" data-value=\"2\">a and c are not related to each other<\/div><div class=\"spq_answer_block\" data-value=\"3\">a, b, and c are all related to each other<\/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: a is related to c<br\/>In a transitive relation, if a is related to b, and b is related to c, then a is related to c.<\/span><\/div><\/div><\/div><div class=\"spq_question_wrapper\" data-answer=\"1\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">3<\/span><h3 class=\"sqp_question_text\">In an anti-transitive relation, if a is related to b, and b is related to c, then<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">a is also related to c<\/div><div class=\"spq_answer_block\" data-value=\"1\">a is not related to c<\/div><div class=\"spq_answer_block\" data-value=\"2\">b is not related to c<\/div><div class=\"spq_answer_block\" data-value=\"3\">b is also related to c<\/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: a is not related to c<br\/>In an anti-transitive relation, if a is related to b, and b is related to c, then a is not related to c.<\/span><\/div><\/div><\/div><div class=\"spq_question_wrapper\" data-answer=\"3\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">4<\/span><h3 class=\"sqp_question_text\">Which of the following is NOT an example of a transitive relation?<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">has the same surname as<\/div><div class=\"spq_answer_block\" data-value=\"1\">Is taller than<\/div><div class=\"spq_answer_block\" data-value=\"2\">is born on the same day as<\/div><div class=\"spq_answer_block\" data-value=\"3\">is a child of<\/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: is a child of<br\/>If a has the same surname as b, and b has the same surname as c, then clearly a has the same first name as c.<br>\r\nIf a is taller than b, and b is taller than c, then clearly a is taller than c.<br>\r\nIf a is born on the same day as b, and b is born on the same day as c, then clearly a is born on the same day as c.<br>\r\nIf a is a child of b, and b is a child of c, then it does not follow that a is a child of c.<\/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\": \"Transitive Relations - Definition, Properties, Examples, FAQs\",        \n        \"about\": {\n                \"@type\": \"Thing\",\n                \"name\": \"Transitive Relations\"\n        },  \n        \"hasPart\": [{\n                    \"@type\": \"Question\",   \n                    \"eduQuestionType\": \"Multiple choice\",\n                    \"learningResourceType\": \"Practice problem\",\n         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is not transitive.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 1,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"\u201cIs equal to\u201d relation on set of real numbers\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"The Is perpendicular to relation on the set of straight lines is not transitive.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 3,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"\u201cIs less than\u201d relation on set of real numbers\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"The Is perpendicular to relation on the set of straight lines is not transitive.\"\n                                    }\n                                }],\n                    \"acceptedAnswer\": {\n                      \"@type\": \"Answer\",\n                      \"position\": 2,\n                      \"encodingFormat\": \"text\/html\",\n                      \"text\": \"\u201cIs perpendicular to\u201d relation on the set of straight lines\",\n                      \"comment\": {\n                          \"@type\": \"Comment\",\n                          \"text\": \"The Is perpendicular to relation on the set of straight lines is not transitive.\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"The Is perpendicular to relation on the set of straight lines is not transitive.\"\n                      }\n                    } \n\n                    },{\n                    \"@type\": \"Question\",   \n                    \"eduQuestionType\": \"Multiple choice\",\n                    \"learningResourceType\": \"Practice problem\",\n                    \"name\": \"In a transitive relation, if a is related to b, and b is related to c, then which of the following is true?\",\n                    \"text\": \"In a transitive relation, if a is related to b, and b is related to c, then which of the following is true?\",\n                    \"comment\": {\n                      \"@type\": \"Comment\",\n                      \"text\": \"In a transitive relation, if a is related to b, and b is related to c, then a is related to c.\"\n                    },\n                    \"encodingFormat\": \"text\/html\",\n                    \"suggestedAnswer\": [ {\n                                \"@type\": \"Answer\",\n                                \"position\": 1,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"c is not related to a\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"In a transitive relation, if a is related to b, and b is related to c, then a is related to c.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 2,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"a and c are not related to each other\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"In a transitive relation, if a is related to b, and b is related to c, then a is related to c.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 3,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"a, b, and c are all related to each other\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"In a transitive relation, if a is related to b, and b is related to c, then a is related to c.\"\n                                    }\n                                }],\n                    \"acceptedAnswer\": {\n                      \"@type\": \"Answer\",\n                      \"position\": 0,\n                      \"encodingFormat\": \"text\/html\",\n                      \"text\": \"a is related to c\",\n                      \"comment\": {\n                          \"@type\": \"Comment\",\n                          \"text\": \"In a transitive relation, if a is related to b, and b is related to c, then a is related to c.\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"In a transitive relation, if a is related to b, and b is related to c, then a is related to c.\"\n                      }\n                    } \n\n                    },{\n                    \"@type\": \"Question\",   \n                    \"eduQuestionType\": \"Multiple choice\",\n                    \"learningResourceType\": \"Practice problem\",\n                    \"name\": \"In an anti-transitive relation, if a is related to b, and b is related to c, then\",\n                    \"text\": \"In an anti-transitive relation, if a is related to b, and b is related to c, then\",\n                    \"comment\": {\n                      \"@type\": \"Comment\",\n                      \"text\": \"In an anti-transitive relation, if a is related to b, and b is related to c, then a is not related to c.\"\n                    },\n                    \"encodingFormat\": \"text\/html\",\n                    \"suggestedAnswer\": [ {\n                                \"@type\": \"Answer\",\n                                \"position\": 0,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"a is also related to c\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"In an anti-transitive relation, if a is related to b, and b is related to c, then a is not related to c.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 2,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"b is not related to c\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"In an anti-transitive relation, if a is related to b, and b is related to c, then a is not related to c.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 3,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"b is also related to c\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"In an anti-transitive relation, if a is related to b, and b is related to c, then a is not related to c.\"\n                                    }\n                                }],\n                    \"acceptedAnswer\": {\n                      \"@type\": \"Answer\",\n                      \"position\": 1,\n                      \"encodingFormat\": \"text\/html\",\n                      \"text\": \"a is not related to c\",\n                      \"comment\": {\n                          \"@type\": \"Comment\",\n                          \"text\": \"In an anti-transitive relation, if a is related to b, and b is related to c, then a is not related to c.\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"In an anti-transitive relation, if a is related to b, and b is related to c, then a is not related to c.\"\n                      }\n                    } \n\n                    },{\n                    \"@type\": \"Question\",   \n                    \"eduQuestionType\": \"Multiple choice\",\n                    \"learningResourceType\": \"Practice problem\",\n                    \"name\": \"Which of the following is NOT an example of a transitive relation?\",\n                    \"text\": \"Which of the following is NOT an example of a transitive relation?\",\n                    \"comment\": {\n                      \"@type\": \"Comment\",\n                      \"text\": \"If a has the same surname as b, and b has the same surname as c, then clearly a has the same first name as c.<br>\r\nIf a is taller than b, and b is taller than c, then clearly a is taller than c.<br>\r\nIf a is born on the same day as b, and b is born on the same day as c, then clearly a is born on the same day as c.<br>\r\nIf a is a child of b, and b is a child of c, then it does not follow that a is a child of c.\"\n                    },\n                    \"encodingFormat\": \"text\/html\",\n                    \"suggestedAnswer\": [ {\n                                \"@type\": \"Answer\",\n                                \"position\": 0,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"has the same surname as\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"If a has the same surname as b, and b has the same surname as c, then clearly a has the same first name as c.<br>\r\nIf a is taller than b, and b is taller than c, then clearly a is taller than c.<br>\r\nIf a is born on the same day as b, and b is born on the same day as c, then clearly a is born on the same day as c.<br>\r\nIf a is a child of b, and b is a child of c, then it does not follow that a is a child of c.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 1,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"Is taller than\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"If a has the same surname as b, and b has the same surname as c, then clearly a has the same first name as c.<br>\r\nIf a is taller than b, and b is taller than c, then clearly a is taller than c.<br>\r\nIf a is born on the same day as b, and b is born on the same day as c, then clearly a is born on the same day as c.<br>\r\nIf a is a child of b, and b is a child of c, then it does not follow that a is a child of c.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 2,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"is born on the same day as\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"If a has the same surname as b, and b has the same surname as c, then clearly a has the same first name as c.<br>\r\nIf a is taller than b, and b is taller than c, then clearly a is taller than c.<br>\r\nIf a is born on the same day as b, and b is born on the same day as c, then clearly a is born on the same day as c.<br>\r\nIf a is a child of b, and b is a child of c, then it does not follow that a is a child of c.\"\n                                    }\n                                }],\n                    \"acceptedAnswer\": {\n                      \"@type\": \"Answer\",\n                      \"position\": 3,\n                      \"encodingFormat\": \"text\/html\",\n                      \"text\": \"is a child of\",\n                      \"comment\": {\n                          \"@type\": \"Comment\",\n                          \"text\": \"If a has the same surname as b, and b has the same surname as c, then clearly a has the same first name as c.<br>\r\nIf a is taller than b, and b is taller than c, then clearly a is taller than c.<br>\r\nIf a is born on the same day as b, and b is born on the same day as c, then clearly a is born on the same day as c.<br>\r\nIf a is a child of b, and b is a child of c, then it does not follow that a is a child of c.\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"If a has the same surname as b, and b has the same surname as c, then clearly a has the same first name as c.<br>\r\nIf a is taller than b, and b is taller than c, then clearly a is taller than c.<br>\r\nIf a is born on the same day as b, and b is born on the same day as c, then clearly a is born on the same day as c.<br>\r\nIf a is a child of b, and b is a child of c, then it does not follow that a is a child of c.\"\n                      }\n                    } \n\n                    }]}<\/script>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"9-frequently-asked-questions-about-transitive-relations\">Frequently Asked Questions about Transitive Relations<\/h2>\n\n\n<div class=\"wp-block-ub-content-toggle\" id=\"ub-content-toggle-3dd73a80-7fc3-49c1-aaab-c2fa6c8a50ba\" 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-0-3dd73a80-7fc3-49c1-aaab-c2fa6c8a50ba\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-3dd73a80-7fc3-49c1-aaab-c2fa6c8a50ba\"><strong>Can a relation be both transitive and 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-0-3dd73a80-7fc3-49c1-aaab-c2fa6c8a50ba\">\n\n<p>Yes, a relation can be both transitive and reflexive. Transitivity and reflexivity are independent properties in math, and a relation can possess both of them simultaneously.<\/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-1-3dd73a80-7fc3-49c1-aaab-c2fa6c8a50ba\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-3dd73a80-7fc3-49c1-aaab-c2fa6c8a50ba\"><strong>How to prove that a relation is transitive?<\/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-1-3dd73a80-7fc3-49c1-aaab-c2fa6c8a50ba\">\n\n<p>To prove that a relation is transitive, we need to prove that for each (a, b) and (b, c) on R, the element (a, c) is also present in 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-2-3dd73a80-7fc3-49c1-aaab-c2fa6c8a50ba\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-3dd73a80-7fc3-49c1-aaab-c2fa6c8a50ba\"><strong>Can a relation R which is transitive be empty?<\/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-2-3dd73a80-7fc3-49c1-aaab-c2fa6c8a50ba\">\n\n<p>Yes, in fact the empty relation is always 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-3-3dd73a80-7fc3-49c1-aaab-c2fa6c8a50ba\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-3dd73a80-7fc3-49c1-aaab-c2fa6c8a50ba\"><strong>What is a non-transitive 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-3-3dd73a80-7fc3-49c1-aaab-c2fa6c8a50ba\">\n\n<p>A relation R is non-transitive if and only if it is neither transitive nor antitransitive. It&#8217;s a relation where there exist elements A, B, and C such that A is related to B, B is related to C, but A is not related to C. It means that the transitive property does not hold for at least one triplet of elements in the relation.<\/p>\n\n<\/div><\/div>\n<\/div>","protected":false},"excerpt":{"rendered":"<p>What Are Transitive Relations? A relation is said to be a transitive relation, if \u201ca is related to b\u201d and \u201cb is related to c\u201d implies that \u201ca is related to c.\u201d In other words, a relation is said to be a transitive relation, if the first element is related to the second element, and &#8230; <a title=\"Transitive Relations &#8211; Definition, Properties, Examples, FAQs\" class=\"read-more\" href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/transitive-relations\" aria-label=\"More on Transitive Relations &#8211; Definition, Properties, 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-34322","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\/34322","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=34322"}],"version-history":[{"count":8,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/posts\/34322\/revisions"}],"predecessor-version":[{"id":39973,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/posts\/34322\/revisions\/39973"}],"wp:attachment":[{"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/media?parent=34322"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/categories?post=34322"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/tags?post=34322"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}