Part 05- Discrete Mathematics for computer science- Anti Symmetric Relation with core Cocept<\/p>\n
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Here is Part 05<\/strong> of Discrete Mathematics for Computer Science<\/em>, focused on the Anti-Symmetric Relation<\/strong>, explained with core concepts, examples, and logic \u2014 especially useful for GATE<\/strong>, CS\/IT<\/strong>, and university-level understanding.<\/p>\n A binary relation RR<\/span>R<\/span><\/span><\/span><\/span><\/strong> on a set AA<\/span>A<\/span><\/span><\/span><\/span> is said to be anti-symmetric<\/strong> if:<\/p>\n If\u00a0(a,b)\u2208R\u00a0and\u00a0(b,a)\u2208R,\u00a0then\u00a0a=b\\text{If } (a, b) \\in R \\text{ and } (b, a) \\in R, \\text{ then } a = b<\/span>If\u00a0<\/span><\/span>(<\/span>a<\/span>,<\/span>b<\/span>)<\/span>\u2208<\/span><\/span>R<\/span>\u00a0and\u00a0<\/span><\/span>(<\/span>b<\/span>,<\/span>a<\/span>)<\/span>\u2208<\/span><\/span>R<\/span>,<\/span>\u00a0then\u00a0<\/span><\/span>a<\/span>=<\/span><\/span>b<\/span><\/span><\/span><\/span><\/span><\/p>\n In simple terms: Let AA<\/span>A<\/span><\/span><\/span><\/span> be a set and R\u2286A\u00d7AR \\subseteq A \\times A<\/span>R<\/span>\u2286<\/span><\/span>A<\/span>\u00d7<\/span><\/span>A<\/span><\/span><\/span><\/span>. \u2200a,b\u2208A,\u00a0(a,b)\u2208R\u2227(b,a)\u2208R\u21d2a=b\\forall a, b \\in A, \\ (a, b) \\in R \\land (b, a) \\in R \\Rightarrow a = b<\/span>\u2200<\/span>a<\/span>,<\/span>b<\/span>\u2208<\/span><\/span>A<\/span>,<\/span>\u00a0<\/span>(<\/span>a<\/span>,<\/span>b<\/span>)<\/span>\u2208<\/span><\/span>R<\/span>\u2227<\/span><\/span>(<\/span>b<\/span>,<\/span>a<\/span>)<\/span>\u2208<\/span><\/span>R<\/span>\u21d2<\/span><\/span>a<\/span>=<\/span><\/span>b<\/span><\/span><\/span><\/span><\/span><\/p>\n Anti-symmetric does not<\/em> mean symmetric = false<\/strong>.<\/p>\n<\/li>\n It is allowed<\/strong> to have both (a,b)(a,b)<\/span>(<\/span>a<\/span>,<\/span>b<\/span>)<\/span><\/span><\/span><\/span> and (b,a)(b,a)<\/span>(<\/span>b<\/span>,<\/span>a<\/span>)<\/span><\/span><\/span><\/span> only when a=ba = b<\/span>a<\/span>=<\/span><\/span>b<\/span><\/span><\/span><\/span><\/strong>.<\/p>\n<\/li>\n Reflexive elements<\/strong> like (a,a)(a,a)<\/span>(<\/span>a<\/span>,<\/span>a<\/span>)<\/span><\/span><\/span><\/span> are fine.<\/p>\n<\/li>\n<\/ul>\n If a\u2264ba \\leq b<\/span>a<\/span>\u2264<\/span><\/span>b<\/span><\/span><\/span><\/span> and b\u2264ab \\leq a<\/span>b<\/span>\u2264<\/span><\/span>a<\/span><\/span><\/span><\/span>, then a=ba = b<\/span>a<\/span>=<\/span><\/span>b<\/span><\/span><\/span><\/span><\/p>\n \u2192 \u2705 Anti-symmetric<\/p>\n If A\u2286BA \\subseteq B<\/span>A<\/span>\u2286<\/span><\/span>B<\/span><\/span><\/span><\/span> and B\u2286AB \\subseteq A<\/span>B<\/span>\u2286<\/span><\/span>A<\/span><\/span><\/span><\/span>, then A=BA = B<\/span>A<\/span>=<\/span><\/span>B<\/span><\/span><\/span><\/span><\/p>\n \u2192 \u2705 Anti-symmetric<\/p>\n Let aRb\u2005\u200a\u27fa\u2005\u200aa\u2223baRb \\iff a \\mid b<\/span>a<\/span>
\n\ud83e\udde0 What is an Anti-Symmetric Relation?<\/strong><\/h2>\n
\ud83d\udc49 If both aa<\/span>a<\/span><\/span><\/span><\/span> is related to bb<\/span>b<\/span><\/span><\/span><\/span><\/strong> and bb<\/span>b<\/span><\/span><\/span><\/span> is related to aa<\/span>a<\/span><\/span><\/span><\/span>**, then they must be the same element<\/strong>.<\/p>\n
\n\ud83d\udcd8 Formal Definition<\/strong><\/h2>\n
RR<\/span>R<\/span><\/span><\/span><\/span> is anti-symmetric<\/strong> if:<\/p>\n
\n\ud83d\udd0d Key Concept:<\/strong><\/h2>\n
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\n\u2705 Examples of Anti-Symmetric Relations<\/strong><\/h2>\n
\ud83d\udd39 Example 1: “Less than or equal to” (\u2264) on real numbers<\/strong><\/h3>\n
\n\ud83d\udd39 Example 2: Subset relation (\u2286) on sets<\/strong><\/h3>\n
\n\ud83d\udd39 Example 3: Divisibility Relation on Natural Numbers<\/strong><\/h3>\n