{"id":3070,"date":"2025-06-06T09:42:34","date_gmt":"2025-06-06T09:42:34","guid":{"rendered":"https:\/\/diznr.com\/?p=3070"},"modified":"2025-06-06T09:42:34","modified_gmt":"2025-06-06T09:42:34","slug":"previous-year-question-papers-gate-for-cse-gate-1996-relations-let-r-be-a-non-relation-empty","status":"publish","type":"post","link":"https:\/\/www.reilsolar.com\/pdf\/previous-year-question-papers-gate-for-cse-gate-1996-relations-let-r-be-a-non-relation-empty\/","title":{"rendered":"previous year question papers gate for cse &#8211; GATE 1996 Relations Let R be a non empty relation."},"content":{"rendered":"<p>previous year question papers gate for cse &#8211; GATE 1996 Relations Let R be a non empty relation.<\/p>\n<p>[fvplayer id=&#8221;237&#8243;]<\/p>\n<p data-start=\"0\" data-end=\"261\">In the GATE 1996 Computer Science exam, there was a question regarding a relation <span class=\"katex\"><span class=\"katex-mathml\">RR<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">R<\/span><\/span><\/span><\/span> defined on a collection of sets, where <span class=\"katex\"><span class=\"katex-mathml\">A\u2009R\u2009BA \\, R \\, B<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">A<\/span><span class=\"mord mathnormal\">R<\/span><span class=\"mord mathnormal\">B<\/span><\/span><\/span><\/span> if and only if <span class=\"katex\"><span class=\"katex-mathml\">A\u2229B=\u2205A \\cap B = \\emptyset<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">A<\/span><span class=\"mbin\">\u2229<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">B<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">\u2205<\/span><\/span><\/span><\/span>. The question asked to identify the correct properties of this relation.<\/p>\n<p data-start=\"263\" data-end=\"300\"><strong data-start=\"263\" data-end=\"300\">Analysis of the Relation <span class=\"katex\"><span class=\"katex-mathml\">RR<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">R<\/span><\/span><\/span><\/span>:<\/strong><\/p>\n<ul data-start=\"302\" data-end=\"1085\">\n<li data-start=\"302\" data-end=\"530\">\n<p data-start=\"304\" data-end=\"530\"><strong data-start=\"304\" data-end=\"320\">Reflexivity:<\/strong> A relation <span class=\"katex\"><span class=\"katex-mathml\">RR<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">R<\/span><\/span><\/span><\/span> is reflexive if every element is related to itself. For any set <span class=\"katex\"><span class=\"katex-mathml\">AA<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">A<\/span><\/span><\/span><\/span>, <span class=\"katex\"><span class=\"katex-mathml\">A\u2229A=AA \\cap A = A<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">A<\/span><span class=\"mbin\">\u2229<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">A<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">A<\/span><\/span><\/span><\/span>, which is generally not empty unless <span class=\"katex\"><span class=\"katex-mathml\">A=\u2205A = \\emptyset<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">A<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">\u2205<\/span><\/span><\/span><\/span>. Therefore, <span class=\"katex\"><span class=\"katex-mathml\">RR<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">R<\/span><\/span><\/span><\/span> is <strong data-start=\"512\" data-end=\"529\">not reflexive<\/strong>.<\/p>\n<\/li>\n<li data-start=\"532\" data-end=\"737\">\n<p data-start=\"534\" data-end=\"737\"><strong data-start=\"534\" data-end=\"547\">Symmetry:<\/strong> A relation <span class=\"katex\"><span class=\"katex-mathml\">RR<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">R<\/span><\/span><\/span><\/span> is symmetric if <span class=\"katex\"><span class=\"katex-mathml\">A\u2009R\u2009BA \\, R \\, B<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">A<\/span><span class=\"mord mathnormal\">R<\/span><span class=\"mord mathnormal\">B<\/span><\/span><\/span><\/span> implies <span class=\"katex\"><span class=\"katex-mathml\">B\u2009R\u2009AB \\, R \\, A<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">B<\/span><span class=\"mord mathnormal\">R<\/span><span class=\"mord mathnormal\">A<\/span><\/span><\/span><\/span>. Given <span class=\"katex\"><span class=\"katex-mathml\">A\u2229B=\u2205A \\cap B = \\emptyset<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">A<\/span><span class=\"mbin\">\u2229<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">B<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">\u2205<\/span><\/span><\/span><\/span>, it follows that <span class=\"katex\"><span class=\"katex-mathml\">B\u2229A=\u2205B \\cap A = \\emptyset<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">B<\/span><span class=\"mbin\">\u2229<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">A<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">\u2205<\/span><\/span><\/span><\/span>. Thus, <span class=\"katex\"><span class=\"katex-mathml\">RR<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">R<\/span><\/span><\/span><\/span> is <strong data-start=\"723\" data-end=\"736\">symmetric<\/strong>.<\/p>\n<\/li>\n<li data-start=\"739\" data-end=\"1085\">\n<p data-start=\"741\" data-end=\"1085\"><strong data-start=\"741\" data-end=\"758\">Transitivity:<\/strong> A relation <span class=\"katex\"><span class=\"katex-mathml\">RR<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">R<\/span><\/span><\/span><\/span> is transitive if <span class=\"katex\"><span class=\"katex-mathml\">A\u2009R\u2009BA \\, R \\, B<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">A<\/span><span class=\"mord mathnormal\">R<\/span><span class=\"mord mathnormal\">B<\/span><\/span><\/span><\/span> and <span class=\"katex\"><span class=\"katex-mathml\">B\u2009R\u2009CB \\, R \\, C<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">B<\/span><span class=\"mord mathnormal\">R<\/span><span class=\"mord mathnormal\">C<\/span><\/span><\/span><\/span> imply <span class=\"katex\"><span class=\"katex-mathml\">A\u2009R\u2009CA \\, R \\, C<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">A<\/span><span class=\"mord mathnormal\">R<\/span><span class=\"mord mathnormal\">C<\/span><\/span><\/span><\/span>. However, consider <span class=\"katex\"><span class=\"katex-mathml\">A={1}A = \\{1\\}<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">A<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mopen\">{<\/span><span class=\"mord\">1<\/span><span class=\"mclose\">}<\/span><\/span><\/span><\/span>, <span class=\"katex\"><span class=\"katex-mathml\">B={2}B = \\{2\\}<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">B<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mopen\">{<\/span><span class=\"mord\">2<\/span><span class=\"mclose\">}<\/span><\/span><\/span><\/span>, and <span class=\"katex\"><span class=\"katex-mathml\">C={1,2}C = \\{1, 2\\}<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">C<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mopen\">{<\/span><span class=\"mord\">1<\/span><span class=\"mpunct\">,<\/span><span class=\"mord\">2<\/span><span class=\"mclose\">}<\/span><\/span><\/span><\/span>. Here, <span class=\"katex\"><span class=\"katex-mathml\">A\u2229B=\u2205A \\cap B = \\emptyset<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">A<\/span><span class=\"mbin\">\u2229<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">B<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">\u2205<\/span><\/span><\/span><\/span> and <span class=\"katex\"><span class=\"katex-mathml\">B\u2229C=\u2205B \\cap C = \\emptyset<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">B<\/span><span class=\"mbin\">\u2229<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">C<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">\u2205<\/span><\/span><\/span><\/span>, but <span class=\"katex\"><span class=\"katex-mathml\">A\u2229C={1}\u2260\u2205A \\cap C = \\{1\\} \\neq \\emptyset<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">A<\/span><span class=\"mbin\">\u2229<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">C<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mopen\">{<\/span><span class=\"mord\">1<\/span><span class=\"mclose\">}<\/span><span class=\"mrel\"><span class=\"mord vbox\"><span class=\"thinbox\"><span class=\"rlap\"><span class=\"inner\"><span class=\"mord\">\ue020<\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><span class=\"base\"><span class=\"mord\">\u2205<\/span><\/span><\/span><\/span>. Therefore, <span class=\"katex\"><span class=\"katex-mathml\">RR<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">R<\/span><\/span><\/span><\/span> is <strong data-start=\"1066\" data-end=\"1084\">not transitive<\/strong>.<\/p>\n<\/li>\n<\/ul>\n<p data-start=\"1087\" data-end=\"1102\"><strong data-start=\"1087\" data-end=\"1102\">Conclusion:<\/strong><\/p>\n<p data-start=\"1104\" data-end=\"1233\">The relation <span class=\"katex\"><span class=\"katex-mathml\">RR<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">R<\/span><\/span><\/span><\/span> is <strong data-start=\"1128\" data-end=\"1141\">symmetric<\/strong> but <strong data-start=\"1146\" data-end=\"1182\">neither reflexive nor transitive<\/strong>. Therefore, it is <strong data-start=\"1201\" data-end=\"1232\">not an equivalence relation<\/strong>.<\/p>\n<p data-start=\"1235\" data-end=\"1287\"><strong data-start=\"1235\" data-end=\"1246\">Answer:<\/strong> <span class=\"katex\"><span class=\"katex-mathml\">RR<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">R<\/span><\/span><\/span><\/span> is symmetric and not transitive.<\/p>\n<p data-start=\"1289\" data-end=\"1370\">For a detailed walkthrough of this problem, you can refer to the following video:<\/p>\n<div class=\"not-prose mb-3 flex flex-col gap-4 text-base\">\n<div>\n<p data-start=\"0\" data-end=\"74\"><span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\">The GATE CSE 1996 exam featured a question on relations, specifically:<\/span><\/p>\n<p data-start=\"76\" data-end=\"89\"><strong data-start=\"76\" data-end=\"89\">Question:<\/strong><\/p>\n<p data-start=\"91\" data-end=\"165\"><span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\">Let <span class=\"katex\"><span class=\"katex-mathml\">RR<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">R<\/span><\/span><\/span><\/span> be a non-empty relation on a collection of sets, defined by <span class=\"katex\"><span class=\"katex-mathml\">A\u2009R\u2009BA\\,R\\,B<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">A<\/span><span class=\"mord mathnormal\">R<\/span><span class=\"mord mathnormal\">B<\/span><\/span><\/span><\/span> if and only if <span class=\"katex\"><span class=\"katex-mathml\">A\u2229B=\u2205A \\cap B = \\emptyset<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">A<\/span><span class=\"mbin\">\u2229<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">B<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">\u2205<\/span><\/span><\/span><\/span>. Then, which of the following statements is true?<\/span><\/p>\n<p data-start=\"167\" data-end=\"179\"><strong data-start=\"167\" data-end=\"179\">Options:<\/strong><\/p>\n<p data-start=\"181\" data-end=\"395\">A. <span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\"><span class=\"katex\"><span class=\"katex-mathml\">RR<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">R<\/span><\/span><\/span><\/span> is reflexive and transitive<\/span><br data-start=\"221\" data-end=\"224\" \/>B. <span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\"><span class=\"katex\"><span class=\"katex-mathml\">RR<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">R<\/span><\/span><\/span><\/span> is symmetric and not transitive<\/span><br data-start=\"266\" data-end=\"269\" \/>C. <span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\"><span class=\"katex\"><span class=\"katex-mathml\">RR<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">R<\/span><\/span><\/span><\/span> is an equivalence relation<\/span><br data-start=\"311\" data-end=\"314\" \/>D. <span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\"><span class=\"katex\"><span class=\"katex-mathml\">RR<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">R<\/span><\/span><\/span><\/span> is not reflexive and not symmetric<\/span><\/p>\n<p data-start=\"397\" data-end=\"487\"><strong data-start=\"397\" data-end=\"408\">Answer:<\/strong> <span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\"><strong data-start=\"0\" data-end=\"46\" data-is-last-node=\"\" data-is-only-node=\"\">B. <span class=\"katex\"><span class=\"katex-mathml\">RR<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">R<\/span><\/span><\/span><\/span> is symmetric and not transitive<\/strong><\/span><\/p>\n<p data-start=\"489\" data-end=\"505\"><strong data-start=\"489\" data-end=\"505\">Explanation:<\/strong><\/p>\n<ul data-start=\"507\" data-end=\"800\">\n<li data-start=\"507\" data-end=\"604\">\n<p data-start=\"509\" data-end=\"604\"><strong data-start=\"509\" data-end=\"525\">Reflexivity:<\/strong> <span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\">A relation <span class=\"katex\"><span class=\"katex-mathml\">RR<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">R<\/span><\/span><\/span><\/span> is reflexive if every element is related to itself. For any set <span class=\"katex\"><span class=\"katex-mathml\">AA<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">A<\/span><\/span><\/span><\/span>, <span class=\"katex\"><span class=\"katex-mathml\">A\u2229A=AA \\cap A = A<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">A<\/span><span class=\"mbin\">\u2229<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">A<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">A<\/span><\/span><\/span><\/span>, which is generally not empty unless <span class=\"katex\"><span class=\"katex-mathml\">A=\u2205A = \\emptyset<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">A<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">\u2205<\/span><\/span><\/span><\/span>. Therefore, <span class=\"katex\"><span class=\"katex-mathml\">RR<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">R<\/span><\/span><\/span><\/span> is not reflexive.<\/span><\/p>\n<\/li>\n<li data-start=\"606\" data-end=\"700\">\n<p data-start=\"608\" data-end=\"700\"><strong data-start=\"608\" data-end=\"621\">Symmetry:<\/strong> <span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\">If <span class=\"katex\"><span class=\"katex-mathml\">A\u2229B=\u2205A \\cap B = \\emptyset<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">A<\/span><span class=\"mbin\">\u2229<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">B<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">\u2205<\/span><\/span><\/span><\/span>, then <span class=\"katex\"><span class=\"katex-mathml\">B\u2229A=\u2205B \\cap A = \\emptyset<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">B<\/span><span class=\"mbin\">\u2229<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">A<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">\u2205<\/span><\/span><\/span><\/span> as well. Hence, if <span class=\"katex\"><span class=\"katex-mathml\">A\u2009R\u2009BA\\,R\\,B<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">A<\/span><span class=\"mord mathnormal\">R<\/span><span class=\"mord mathnormal\">B<\/span><\/span><\/span><\/span>, then <span class=\"katex\"><span class=\"katex-mathml\">B\u2009R\u2009AB\\,R\\,A<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">B<\/span><span class=\"mord mathnormal\">R<\/span><span class=\"mord mathnormal\">A<\/span><\/span><\/span><\/span>; thus, <span class=\"katex\"><span class=\"katex-mathml\">RR<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">R<\/span><\/span><\/span><\/span> is symmetric.<\/span><\/p>\n<\/li>\n<li data-start=\"702\" data-end=\"800\">\n<p data-start=\"704\" data-end=\"800\"><strong data-start=\"704\" data-end=\"721\">Transitivity:<\/strong> <span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\">Consider sets <span class=\"katex\"><span class=\"katex-mathml\">A={1}A = \\{1\\}<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">A<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mopen\">{<\/span><span class=\"mord\">1<\/span><span class=\"mclose\">}<\/span><\/span><\/span><\/span>, <span class=\"katex\"><span class=\"katex-mathml\">B={2}B = \\{2\\}<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">B<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mopen\">{<\/span><span class=\"mord\">2<\/span><span class=\"mclose\">}<\/span><\/span><\/span><\/span>, and <span class=\"katex\"><span class=\"katex-mathml\">C={1,2}C = \\{1,2\\}<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">C<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mopen\">{<\/span><span class=\"mord\">1<\/span><span class=\"mpunct\">,<\/span><span class=\"mord\">2<\/span><span class=\"mclose\">}<\/span><\/span><\/span><\/span>. We have <span class=\"katex\"><span class=\"katex-mathml\">A\u2229B=\u2205A \\cap B = \\emptyset<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">A<\/span><span class=\"mbin\">\u2229<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">B<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">\u2205<\/span><\/span><\/span><\/span> and <span class=\"katex\"><span class=\"katex-mathml\">B\u2229C=\u2205B \\cap C = \\emptyset<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">B<\/span><span class=\"mbin\">\u2229<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">C<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">\u2205<\/span><\/span><\/span><\/span>, so <span class=\"katex\"><span class=\"katex-mathml\">A\u2009R\u2009BA\\,R\\,B<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">A<\/span><span class=\"mord mathnormal\">R<\/span><span class=\"mord mathnormal\">B<\/span><\/span><\/span><\/span> and <span class=\"katex\"><span class=\"katex-mathml\">B\u2009R\u2009CB\\,R\\,C<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">B<\/span><span class=\"mord mathnormal\">R<\/span><span class=\"mord mathnormal\">C<\/span><\/span><\/span><\/span>. However, <span class=\"katex\"><span class=\"katex-mathml\">A\u2229C={1}\u2260\u2205A \\cap C = \\{1\\} \\neq \\emptyset<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">A<\/span><span class=\"mbin\">\u2229<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">C<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mopen\">{<\/span><span class=\"mord\">1<\/span><span class=\"mclose\">}<\/span><span class=\"mrel\"><span class=\"mord vbox\"><span class=\"thinbox\"><span class=\"rlap\"><span class=\"inner\"><span class=\"mord\">\ue020<\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><span class=\"base\"><span class=\"mord\">\u2205<\/span><\/span><\/span><\/span>, so <span class=\"katex\"><span class=\"katex-mathml\">A\u2009R\u2009CA\\,R\\,C<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">A<\/span><span class=\"mord mathnormal\">R<\/span><span class=\"mord mathnormal\">C<\/span><\/span><\/span><\/span> does not hold. Therefore, <span class=\"katex\"><span class=\"katex-mathml\">RR<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">R<\/span><\/span><\/span><\/span> is not transitive.<\/span><\/p>\n<\/li>\n<\/ul>\n<p data-start=\"802\" data-end=\"880\"><span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\">This analysis confirms that option B is correct.<\/span><\/p>\n<p data-start=\"882\" data-end=\"968\">For a detailed walkthrough of this question, you can refer to the following resources:<\/p>\n<ul data-start=\"970\" data-end=\"1175\">\n<li data-start=\"970\" data-end=\"1073\">\n<p data-start=\"972\" data-end=\"1073\"><a class=\"cursor-pointer\" target=\"_new\" rel=\"noopener\" data-start=\"972\" data-end=\"1073\">GATE CSE 1996 Question 2.2 on GateOverflow<\/a><\/p>\n<\/li>\n<li data-start=\"1074\" data-end=\"1175\">\n<p data-start=\"1076\" data-end=\"1175\">Set Theory &#8211; GATE CSE 1996 Solved Question &#8211; YouTube<\/p>\n<\/li>\n<\/ul>\n<p data-start=\"1177\" data-end=\"1260\">If you need further assistance or explanations on related topics, feel free to ask!<\/p>\n<h3 data-start=\"1177\" data-end=\"1260\"><a href=\"https:\/\/gateforum.com\/wp-content\/uploads\/2013\/01\/CS-1996.pdf\" target=\"_blank\" rel=\"noopener\">previous year question papers gate for cse &#8211; GATE 1996 Relations Let R be a non empty relation.<\/a><\/h3>\n<h3 class=\"LC20lb MBeuO DKV0Md\"><a href=\"https:\/\/gate.iitk.ac.in\/GATE2023\/doc\/papers\/2012\/cs_2012.pdf\" target=\"_blank\" rel=\"noopener\">cs : computer science &amp; information technology<\/a><\/h3>\n<h3 class=\"LC20lb MBeuO DKV0Md\"><a href=\"https:\/\/www.madeeasy.in\/uploads\/examsolution\/209ufrep_CS-GATE-2020-Solution.pdf\" target=\"_blank\" rel=\"noopener\">GATE 2025 Download<\/a><\/h3>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>previous year question papers gate for cse &#8211; GATE 1996 Relations Let R be a non empty relation. [fvplayer id=&#8221;237&#8243;] In the GATE 1996 Computer Science exam, there was a question regarding a relation RRR defined on a collection of sets, where A\u2009R\u2009BA \\, R \\, BARB if and only if A\u2229B=\u2205A \\cap B = [&hellip;]<\/p>\n","protected":false},"author":71,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[76],"tags":[],"class_list":["post-3070","post","type-post","status-publish","format-standard","hentry","category-discrete-mathematics"],"_links":{"self":[{"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/posts\/3070","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/users\/71"}],"replies":[{"embeddable":true,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/comments?post=3070"}],"version-history":[{"count":0,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/posts\/3070\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/media?parent=3070"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/categories?post=3070"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/tags?post=3070"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}