previous year question papers gate- GATE 1995-relation-Let R be a symmetric and transitive relation.<\/p>\n
[fvplayer id=”238″]<\/p>\n
Question:<\/strong> (A) R is reflexive<\/strong> We are given that R is symmetric and transitive<\/strong>. Let’s analyze the given options:<\/p>\n Option (D) R is reflexive or empty<\/strong><\/p>\n This is a common GATE question<\/strong> based on properties of relations<\/strong>. Would you like more practice questions<\/strong> on this topic?<\/p>\n Here’s a detailed explanation of the GATE 1995 question<\/strong> based on Relations<\/strong> \u2014 specifically focusing on Symmetric<\/strong> and Transitive<\/strong> relations.<\/p>\n Let RR<\/span> be a symmetric<\/strong> and transitive<\/strong> relation on a set AA<\/span>. Suppose (a,b)\u2208R(a, b) \\in R<\/span>. Options (typical structure):<\/strong> We are given:<\/p>\n And we are told: D) All of the above<\/strong><\/p>\n
Let R<\/strong> be a symmetric<\/strong> and transitive<\/strong> relation on a set A<\/strong>. Then which of the following is always true<\/strong>?<\/p>\n
(B) R is an equivalence relation<\/strong>
(C) R is anti-symmetric<\/strong>
(D) R is reflexive or empty<\/strong><\/p>\n\u00a0Step-by-Step Solution<\/strong><\/h3>\n
\u00a01. Reflexivity Check<\/strong><\/h3>\n
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\u00a0So, R is NOT necessarily reflexive<\/strong> \u2192 (Option A is False).<\/strong><\/li>\n<\/ul>\n\u00a02. Equivalence Relation Check<\/strong><\/h3>\n
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Option B is False.<\/strong><\/li>\n<\/ul>\n\u00a03. Anti-Symmetry Check<\/strong><\/h3>\n
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Option C is False.<\/strong><\/li>\n<\/ul>\n\u00a04. Reflexive or Empty Check<\/strong><\/h3>\n
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R is reflexive<\/strong> (if it contains all (a, a) pairs).
R is empty<\/strong> (which still satisfies symmetry and transitivity).
Option D is True<\/strong> (Correct Answer).<\/li>\n<\/ul>\n\u00a0Final Answer:<\/strong><\/h3>\n
previous year question papers gate- GATE 1995-relation-Let R be a symmetric and transitive relation.<\/a><\/h3>\n
GATE CS – 1995<\/a><\/h3>\n
\n\ud83e\udde0 GATE 1995 \u2013 Set Theory \/ Relations Question<\/strong><\/h2>\n
\u2753Question:<\/strong><\/h3>\n
\nThen which of the following must also be true?<\/p>\n
\nA) (b,a)\u2208R(b, a) \\in R<\/span>
\nB) (a,a)\u2208R(a, a) \\in R<\/span>
\nC) (b,b)\u2208R(b, b) \\in R<\/span>
\nD) All of the above<\/p>\n
\n\u2705 Solution with Concept:<\/strong><\/h2>\n
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\nIf (a,b)\u2208R\u21d2(b,a)\u2208R(a, b) \\in R \\Rightarrow (b, a) \\in R<\/span><\/li>\n
\nIf (a,b)\u2208R(a, b) \\in R<\/span> and (b,c)\u2208R\u21d2(a,c)\u2208R(b, c) \\in R \\Rightarrow (a, c) \\in R<\/span><\/li>\n<\/ul>\n
\n(a,b)\u2208R(a, b) \\in R<\/span><\/p>\n
\n\ud83d\udd0d Step-by-step Reasoning:<\/h3>\n
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\nSince (a,b)\u2208R(a, b) \\in R<\/span>, \u21d2 (b,a)\u2208R(b, a) \\in R<\/span><\/li>\n\n
\n\u21d2 By transitivity: (a,a)\u2208R(a, a) \\in R<\/span> \u2705<\/li>\n<\/ul>\n<\/li>\n\n
\n\u21d2 (b,b)\u2208R(b, b) \\in R<\/span> \u2705<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n
\n\u2705 Final Answer:<\/h3>\n
\n\ud83d\udccc GATE Concept Summary:<\/h3>\n