{"id":3020,"date":"2025-06-07T14:59:41","date_gmt":"2025-06-07T14:59:41","guid":{"rendered":"https:\/\/diznr.com\/?p=3020"},"modified":"2025-06-07T14:59:41","modified_gmt":"2025-06-07T14:59:41","slug":"day-03part-12-discrete-mathematics-examples-based-on-lattice-for-finding-lattice-in-speed-faster","status":"publish","type":"post","link":"https:\/\/www.reilsolar.com\/pdf\/day-03part-12-discrete-mathematics-examples-based-on-lattice-for-finding-lattice-in-speed-faster\/","title":{"rendered":"Day 03Part 12- Discrete mathematics – Examples based on lattice for finding lattice in faster speed"},"content":{"rendered":"

Day 03Part 12- Discrete mathematics – Examples based on lattice for finding lattice in faster speed<\/p>\n

[fvplayer id=”219″]<\/p>\n

To efficiently determine whether a partially ordered set (poset) is a lattice<\/strong>, especially in the context of GATE or other competitive exams, you can follow a systematic approach.<\/span> Here’s a step-by-step guide with examples to help you quickly assess lattice structures.<\/p>\n

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\u2705 Quick Method to Identify a Lattice<\/strong><\/h2>\n

A lattice<\/strong> is a poset in which every pair of elements<\/strong> has both:<\/span><\/p>\n

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    a least upper bound (LUB)<\/strong>, also known as join<\/strong><\/span><\/p>\n<\/li>\n

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    a greatest lower bound (GLB)<\/strong>, also known as meet<\/strong><\/span><\/p>\n<\/li>\n<\/ul>\n

    To verify if a poset is a lattice:<\/span>XYQuadrat<\/span><\/span><\/span><\/span><\/p>\n

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      List all pairs of elements<\/strong> in the poset.<\/p>\n<\/li>\n

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      For each pair, determine:<\/p>\n

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        The LUB<\/strong>: the smallest element greater than or equal to both.<\/span><\/p>\n<\/li>\n

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        The GLB<\/strong>: the largest element less than or equal to both.<\/span><\/p>\n<\/li>\n<\/ul>\n<\/li>\n

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        If both LUB and GLB exist for every pair<\/strong>, the poset is a lattice.<\/span><\/p>\n<\/li>\n<\/ol>\n

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        \ud83e\uddea Example 1:<\/strong><\/h2>\n

        Set:<\/strong> {1,3,6,9,12}\\{1, 3, 6, 9, 12\\}<\/span>{<\/span>1<\/span>,<\/span>3<\/span>,<\/span>6<\/span>,<\/span>9<\/span>,<\/span>12<\/span>}<\/span><\/span><\/span><\/span> with the relation “divides” (denoted as \u2223|<\/span>\u2223<\/span><\/span><\/span><\/span>).<\/span><\/p>\n

        Analysis:<\/strong><\/p>\n

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          Pair (3, 6):<\/strong><\/p>\n

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            GLB: gcd\u2061(3,6)=3\\gcd(3, 6) = 3<\/span>gcd<\/span>(<\/span>3<\/span>,<\/span>6<\/span>)<\/span>=<\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/p>\n<\/li>\n

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            LUB: lcm(3,6)=6\\text{lcm}(3, 6) = 6<\/span>lcm<\/span><\/span>(<\/span>3<\/span>,<\/span>6<\/span>)<\/span>=<\/span><\/span>6<\/span><\/span><\/span><\/span><\/span><\/p>\n<\/li>\n<\/ul>\n<\/li>\n

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            Pair (6, 9):<\/strong><\/p>\n

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              GLB: gcd\u2061(6,9)=3\\gcd(6, 9) = 3<\/span>gcd<\/span>(<\/span>6<\/span>,<\/span>9<\/span>)<\/span>=<\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/p>\n<\/li>\n

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              LUB: lcm(6,9)=18\\text{lcm}(6, 9) = 18<\/span>lcm<\/span><\/span>(<\/span>6<\/span>,<\/span>9<\/span>)<\/span>=<\/span><\/span>18<\/span><\/span><\/span><\/span><\/span><\/p>\n<\/li>\n

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              However, 18 is not in the set.<\/span><\/p>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n

              Conclusion:<\/strong> Since the LUB for (6, 9) is not in the set, this poset is not<\/strong> a lattice.<\/span><\/p>\n

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              \ud83e\uddea Example 2:<\/strong><\/h2>\n

              Set:<\/strong> {1,2,4,8,16}\\{1, 2, 4, 8, 16\\}<\/span>{<\/span>1<\/span>,<\/span>2<\/span>,<\/span>4<\/span>,<\/span>8<\/span>,<\/span>16<\/span>}<\/span><\/span><\/span><\/span> with the relation “divides” (\u2223|<\/span>\u2223<\/span><\/span><\/span><\/span>).<\/span><\/p>\n

              Analysis:<\/strong><\/p>\n

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                For any pair, both gcd\u2061\\gcd<\/span>gcd<\/span><\/span><\/span><\/span> and lcm\\text{lcm}<\/span>lcm<\/span><\/span><\/span><\/span><\/span> are within the set.<\/span><\/p>\n<\/li>\n<\/ul>\n

                Conclusion:<\/strong> This poset is<\/strong> a lattice.<\/span><\/p>\n

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                \ud83e\uddea Example 3:<\/strong><\/h2>\n

                Set:<\/strong> {1,2,3,4,6,12}\\{1, 2, 3, 4, 6, 12\\}<\/span>{<\/span>1<\/span>,<\/span>2<\/span>,<\/span>3<\/span>,<\/span>4<\/span>,<\/span>6<\/span>,<\/span>12<\/span>}<\/span><\/span><\/span><\/span> with the relation “divides” (\u2223|<\/span>\u2223<\/span><\/span><\/span><\/span>).<\/span><\/p>\n

                Analysis:<\/strong><\/p>\n

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                  Pair (3, 4):<\/strong><\/p>\n

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                    GLB: gcd\u2061(3,4)=1\\gcd(3, 4) = 1<\/span>gcd<\/span>(<\/span>3<\/span>,<\/span>4<\/span>)<\/span>=<\/span><\/span>1<\/span><\/span><\/span><\/span><\/span><\/p>\n<\/li>\n

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                    LUB: lcm(3,4)=12\\text{lcm}(3, 4) = 12<\/span>lcm<\/span><\/span>(<\/span>3<\/span>,<\/span>4<\/span>)<\/span>=<\/span><\/span>12<\/span><\/span><\/span><\/span><\/span><\/p>\n<\/li>\n<\/ul>\n<\/li>\n

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                    All other pairs also have their gcd\u2061\\gcd<\/span>gcd<\/span><\/span><\/span><\/span> and lcm\\text{lcm}<\/span>lcm<\/span><\/span><\/span><\/span><\/span> within the set.<\/span><\/p>\n<\/li>\n<\/ul>\n

                    Conclusion:<\/strong> This poset is<\/strong> a lattice.<\/span><\/p>\n

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                    \ud83e\udde0 Tips for Faster Identification:<\/strong><\/h2>\n
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                      Use Hasse Diagrams:<\/strong> Visual representations can help quickly identify LUBs and GLBs.<\/span><\/p>\n<\/li>\n

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                      Check for Completeness:<\/strong> Ensure that for every pair, both LUB and GLB exist within the set.<\/span><\/p>\n<\/li>\n

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                      Utilize Mathematical Properties:<\/strong> For numerical sets, leveraging gcd\u2061\\gcd<\/span>gcd<\/span><\/span><\/span><\/span> and lcm\\text{lcm}<\/span>lcm<\/span><\/span><\/span><\/span><\/span> can expedite the process.<\/span><\/p>\n<\/li>\n<\/ul>\n

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                      \ud83c\udfa5 Recommended Video Tutorials:<\/strong><\/h2>\n
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                        Problems on Lattice – Poset and Lattice – Discrete Mathematics:<\/strong> This video covers various problems related to lattices and posets, providing clear explanations and examples.<\/span><\/p>\n<\/li>\n

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                        Check if a poset is a lattice:<\/strong> This tutorial demonstrates how to determine whether a given poset is a lattice, including checks for distributive lattices.<\/span><\/p>\n<\/li>\n<\/ul>\n

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                        If you need further assistance with specific examples or concepts related to lattices, feel free to ask!<\/p>\n","protected":false},"excerpt":{"rendered":"

                        Day 03Part 12- Discrete mathematics – Examples based on lattice for finding lattice in faster speed [fvplayer id=”219″] To efficiently determine whether a partially ordered set (poset) is a lattice, especially in the context of GATE or other competitive exams, you can follow a systematic approach. Here’s a step-by-step guide with examples to help you […]<\/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-3020","post","type-post","status-publish","format-standard","hentry","category-discrete-mathematics"],"_links":{"self":[{"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/posts\/3020","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=3020"}],"version-history":[{"count":0,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/posts\/3020\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/media?parent=3020"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/categories?post=3020"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/tags?post=3020"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}