{"id":3013,"date":"2025-06-07T14:52:59","date_gmt":"2025-06-07T14:52:59","guid":{"rendered":"https:\/\/diznr.com\/?p=3013"},"modified":"2025-06-07T14:52:59","modified_gmt":"2025-06-07T14:52:59","slug":"day-03part-15-discrete-mathematics-for-computer-science-properties-of-lattices-example-with","status":"publish","type":"post","link":"https:\/\/www.reilsolar.com\/pdf\/day-03part-15-discrete-mathematics-for-computer-science-properties-of-lattices-example-with\/","title":{"rendered":"Day 03Part 15-Discrete mathematics for computer science-Properties of lattices with example."},"content":{"rendered":"<p>Day 03Part 15-Discrete mathematics for computer science-Properties of lattices with example.<\/p>\n<p>[fvplayer id=&#8221;216&#8243;]<\/p>\n<p class=\"\" 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\">Here is a concise overview of <strong data-start=\"30\" data-end=\"42\">lattices<\/strong> in discrete mathematics, focusing on their properties with examples, suitable for GATE CSE\/IT preparation:<\/span><\/p>\n<hr class=\"\" data-start=\"76\" data-end=\"79\" \/>\n<h2 class=\"\" data-start=\"81\" data-end=\"109\">\ud83d\udcd8 <strong data-start=\"87\" data-end=\"109\">What is a Lattice?<\/strong><\/h2>\n<p class=\"\" data-start=\"111\" data-end=\"185\"><span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\">A <strong data-start=\"2\" data-end=\"13\">lattice<\/strong> is a partially ordered set (POSET) in which every pair of elements has both:<\/span><\/p>\n<ul data-start=\"187\" data-end=\"354\">\n<li class=\"\" data-start=\"187\" data-end=\"268\">\n<p class=\"\" data-start=\"189\" data-end=\"268\"><strong data-start=\"189\" data-end=\"216\">Least Upper Bound (LUB)<\/strong>, also known as <strong data-start=\"232\" data-end=\"240\">join<\/strong> (denoted by <span class=\"katex\"><span class=\"katex-mathml\">a\u2228ba \\lor b<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">a<\/span><span class=\"mbin\">\u2228<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">b<\/span><\/span><\/span><\/span>)<\/p>\n<\/li>\n<li class=\"\" data-start=\"269\" data-end=\"354\">\n<p class=\"\" data-start=\"271\" data-end=\"354\"><strong data-start=\"271\" data-end=\"301\">Greatest Lower Bound (GLB)<\/strong>, also known as <strong data-start=\"317\" data-end=\"325\">meet<\/strong> (denoted by <span class=\"katex\"><span class=\"katex-mathml\">a\u2227ba \\land b<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">a<\/span><span class=\"mbin\">\u2227<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">b<\/span><\/span><\/span><\/span>)<\/p>\n<\/li>\n<\/ul>\n<p class=\"\" data-start=\"356\" data-end=\"472\"><span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\">This structure ensures that for any two elements in the set, both their join and meet exist.<\/span><\/p>\n<hr class=\"\" data-start=\"474\" data-end=\"477\" \/>\n<h2 class=\"\" data-start=\"479\" data-end=\"515\">\ud83d\udd0d <strong data-start=\"485\" data-end=\"515\">Key Properties of Lattices<\/strong><\/h2>\n<ol data-start=\"517\" data-end=\"900\">\n<li class=\"\" data-start=\"517\" data-end=\"605\">\n<p class=\"\" data-start=\"520\" data-end=\"541\"><strong data-start=\"520\" data-end=\"540\">Commutative Laws<\/strong>:<\/p>\n<ul data-start=\"545\" data-end=\"605\">\n<li class=\"\" data-start=\"545\" data-end=\"572\">\n<p class=\"\" data-start=\"547\" data-end=\"572\"><span class=\"katex\"><span class=\"katex-mathml\">a\u2228b=b\u2228aa \\lor b = b \\lor a<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">a<\/span><span class=\"mbin\">\u2228<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">b<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">b<\/span><span class=\"mbin\">\u2228<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">a<\/span><\/span><\/span><\/span><\/p>\n<\/li>\n<li class=\"\" data-start=\"576\" data-end=\"605\">\n<p class=\"\" data-start=\"578\" data-end=\"605\"><span class=\"katex\"><span class=\"katex-mathml\">a\u2227b=b\u2227aa \\land b = b \\land a<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">a<\/span><span class=\"mbin\">\u2227<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">b<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">b<\/span><span class=\"mbin\">\u2227<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">a<\/span><\/span><\/span><\/span><\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li class=\"\" data-start=\"607\" data-end=\"733\">\n<p class=\"\" data-start=\"610\" data-end=\"631\"><strong data-start=\"610\" data-end=\"630\">Associative Laws<\/strong>:<\/p>\n<ul data-start=\"635\" data-end=\"733\">\n<li class=\"\" data-start=\"635\" data-end=\"680\">\n<p class=\"\" data-start=\"637\" data-end=\"680\"><span class=\"katex\"><span class=\"katex-mathml\">a\u2228(b\u2228c)=(a\u2228b)\u2228ca \\lor (b \\lor c) = (a \\lor b) \\lor c<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">a<\/span><span class=\"mbin\">\u2228<\/span><\/span><span class=\"base\"><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">b<\/span><span class=\"mbin\">\u2228<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">c<\/span><span class=\"mclose\">)<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">a<\/span><span class=\"mbin\">\u2228<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">b<\/span><span class=\"mclose\">)<\/span><span class=\"mbin\">\u2228<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">c<\/span><\/span><\/span><\/span><\/p>\n<\/li>\n<li class=\"\" data-start=\"684\" data-end=\"733\">\n<p class=\"\" data-start=\"686\" data-end=\"733\"><span class=\"katex\"><span class=\"katex-mathml\">a\u2227(b\u2227c)=(a\u2227b)\u2227ca \\land (b \\land c) = (a \\land b) \\land c<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">a<\/span><span class=\"mbin\">\u2227<\/span><\/span><span class=\"base\"><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">b<\/span><span class=\"mbin\">\u2227<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">c<\/span><span class=\"mclose\">)<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">a<\/span><span class=\"mbin\">\u2227<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">b<\/span><span class=\"mclose\">)<\/span><span class=\"mbin\">\u2227<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">c<\/span><\/span><\/span><\/span><\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li class=\"\" data-start=\"735\" data-end=\"826\">\n<p class=\"\" data-start=\"738\" data-end=\"758\"><strong data-start=\"738\" data-end=\"757\">Absorption Laws<\/strong>:<\/p>\n<ul data-start=\"762\" data-end=\"826\">\n<li class=\"\" data-start=\"762\" data-end=\"792\">\n<p class=\"\" data-start=\"764\" data-end=\"792\"><span class=\"katex\"><span class=\"katex-mathml\">a\u2228(a\u2227b)=aa \\lor (a \\land b) = a<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">a<\/span><span class=\"mbin\">\u2228<\/span><\/span><span class=\"base\"><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">a<\/span><span class=\"mbin\">\u2227<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">b<\/span><span class=\"mclose\">)<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">a<\/span><\/span><\/span><\/span><\/p>\n<\/li>\n<li class=\"\" data-start=\"796\" data-end=\"826\">\n<p class=\"\" data-start=\"798\" data-end=\"826\"><span class=\"katex\"><span class=\"katex-mathml\">a\u2227(a\u2228b)=aa \\land (a \\lor b) = a<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">a<\/span><span class=\"mbin\">\u2227<\/span><\/span><span class=\"base\"><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">a<\/span><span class=\"mbin\">\u2228<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">b<\/span><span class=\"mclose\">)<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">a<\/span><\/span><\/span><\/span><\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li class=\"\" data-start=\"828\" data-end=\"900\">\n<p class=\"\" data-start=\"831\" data-end=\"851\"><strong data-start=\"831\" data-end=\"850\">Idempotent Laws<\/strong>:<\/p>\n<ul data-start=\"855\" data-end=\"900\">\n<li class=\"\" data-start=\"855\" data-end=\"875\">\n<p class=\"\" data-start=\"857\" data-end=\"875\"><span class=\"katex\"><span class=\"katex-mathml\">a\u2228a=aa \\lor a = a<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">a<\/span><span class=\"mbin\">\u2228<\/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><\/p>\n<\/li>\n<li class=\"\" data-start=\"879\" data-end=\"900\">\n<p class=\"\" data-start=\"881\" data-end=\"900\"><span class=\"katex\"><span class=\"katex-mathml\">a\u2227a=aa \\land a = a<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">a<\/span><span class=\"mbin\">\u2227<\/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><\/p>\n<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<p class=\"\" data-start=\"902\" data-end=\"1020\"><span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\">These properties define the algebraic structure of lattices and are fundamental in their analysis.<\/span><\/p>\n<hr class=\"\" data-start=\"1022\" data-end=\"1025\" \/>\n<h2 class=\"\" data-start=\"1027\" data-end=\"1054\">\ud83e\uddf1 <strong data-start=\"1033\" data-end=\"1054\">Types of Lattices<\/strong><\/h2>\n<ol data-start=\"1056\" data-end=\"1964\">\n<li class=\"\" data-start=\"1056\" data-end=\"1208\">\n<p class=\"\" data-start=\"1059\" data-end=\"1079\"><strong data-start=\"1059\" data-end=\"1078\">Bounded Lattice<\/strong>:<\/p>\n<ul data-start=\"1083\" data-end=\"1208\">\n<li class=\"\" data-start=\"1083\" data-end=\"1124\">\n<p class=\"\" data-start=\"1085\" data-end=\"1124\"><span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\">Contains both a <strong data-start=\"16\" data-end=\"37\">least element (0)<\/strong> and a <strong data-start=\"44\" data-end=\"68\">greatest element (1)<\/strong>.<\/span><\/p>\n<\/li>\n<li class=\"\" data-start=\"1128\" data-end=\"1208\">\n<p class=\"\" data-start=\"1130\" data-end=\"1208\"><span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\">For all <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> in the lattice: <span class=\"katex\"><span class=\"katex-mathml\">0\u2264a\u226410 \\leq a \\leq 1<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">0<\/span><span class=\"mrel\">\u2264<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">a<\/span><span class=\"mrel\">\u2264<\/span><\/span><span class=\"base\"><span class=\"mord\">1<\/span><\/span><\/span><\/span>.<\/span><\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li class=\"\" data-start=\"1210\" data-end=\"1532\">\n<p class=\"\" data-start=\"1213\" data-end=\"1238\"><strong data-start=\"1213\" data-end=\"1237\">Distributive Lattice<\/strong>:<\/p>\n<ul data-start=\"1242\" data-end=\"1532\">\n<li class=\"\" data-start=\"1242\" data-end=\"1408\">\n<p class=\"\" data-start=\"1244\" data-end=\"1283\"><span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\">Satisfies the distributive laws:<\/span><\/p>\n<ul data-start=\"1289\" data-end=\"1408\">\n<li class=\"\" data-start=\"1289\" data-end=\"1345\">\n<p class=\"\" data-start=\"1291\" data-end=\"1345\"><span class=\"katex\"><span class=\"katex-mathml\">a\u2228(b\u2227c)=(a\u2228b)\u2227(a\u2228c)a \\lor (b \\land c) = (a \\lor b) \\land (a \\lor c)<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">a<\/span><span class=\"mbin\">\u2228<\/span><\/span><span class=\"base\"><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">b<\/span><span class=\"mbin\">\u2227<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">c<\/span><span class=\"mclose\">)<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">a<\/span><span class=\"mbin\">\u2228<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">b<\/span><span class=\"mclose\">)<\/span><span class=\"mbin\">\u2227<\/span><\/span><span class=\"base\"><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">a<\/span><span class=\"mbin\">\u2228<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">c<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/p>\n<\/li>\n<li class=\"\" data-start=\"1351\" data-end=\"1408\">\n<p class=\"\" data-start=\"1353\" data-end=\"1408\"><span class=\"katex\"><span class=\"katex-mathml\">a\u2227(b\u2228c)=(a\u2227b)\u2228(a\u2227c)a \\land (b \\lor c) = (a \\land b) \\lor (a \\land c)<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">a<\/span><span class=\"mbin\">\u2227<\/span><\/span><span class=\"base\"><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">b<\/span><span class=\"mbin\">\u2228<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">c<\/span><span class=\"mclose\">)<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">a<\/span><span class=\"mbin\">\u2227<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">b<\/span><span class=\"mclose\">)<\/span><span class=\"mbin\">\u2228<\/span><\/span><span class=\"base\"><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">a<\/span><span class=\"mbin\">\u2227<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">c<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li class=\"\" data-start=\"1412\" data-end=\"1532\">\n<p class=\"\" data-start=\"1414\" data-end=\"1532\"><span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\">A lattice is distributive if it does not contain sublattices isomorphic to <span class=\"katex\"><span class=\"katex-mathml\">M3M_3<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\"><span class=\"mord mathnormal\">M<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">3<\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> or <span class=\"katex\"><span class=\"katex-mathml\">N5N_5<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\"><span class=\"mord mathnormal\">N<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">5<\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>.<\/span><\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li class=\"\" data-start=\"1534\" data-end=\"1641\">\n<p class=\"\" data-start=\"1537\" data-end=\"1557\"><strong data-start=\"1537\" data-end=\"1556\">Modular Lattice<\/strong>:<\/p>\n<ul data-start=\"1561\" data-end=\"1641\">\n<li class=\"\" data-start=\"1561\" data-end=\"1641\">\n<p class=\"\" data-start=\"1563\" data-end=\"1641\"><span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\">Satisfies the modular law: If <span class=\"katex\"><span class=\"katex-mathml\">a\u2264ca \\leq c<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">a<\/span><span class=\"mrel\">\u2264<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">c<\/span><\/span><\/span><\/span>, then <span class=\"katex\"><span class=\"katex-mathml\">a\u2228(b\u2227c)=(a\u2228b)\u2227ca \\lor (b \\land c) = (a \\lor b) \\land c<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">a<\/span><span class=\"mbin\">\u2228<\/span><\/span><span class=\"base\"><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">b<\/span><span class=\"mbin\">\u2227<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">c<\/span><span class=\"mclose\">)<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">a<\/span><span class=\"mbin\">\u2228<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">b<\/span><span class=\"mclose\">)<\/span><span class=\"mbin\">\u2227<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">c<\/span><\/span><\/span><\/span>.<\/span><\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li class=\"\" data-start=\"1643\" data-end=\"1769\">\n<p class=\"\" data-start=\"1646\" data-end=\"1671\"><strong data-start=\"1646\" data-end=\"1670\">Complemented Lattice<\/strong>:<\/p>\n<ul data-start=\"1675\" data-end=\"1769\">\n<li class=\"\" data-start=\"1675\" data-end=\"1769\">\n<p class=\"\" data-start=\"1677\" data-end=\"1716\"><span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\">A bounded lattice where every element <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> has a complement <span class=\"katex\"><span class=\"katex-mathml\">bb<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">b<\/span><\/span><\/span><\/span> such that:<\/span><\/p>\n<ul data-start=\"1722\" data-end=\"1769\">\n<li class=\"\" data-start=\"1722\" data-end=\"1742\">\n<p class=\"\" data-start=\"1724\" data-end=\"1742\"><span class=\"katex\"><span class=\"katex-mathml\">a\u2228b=1a \\lor b = 1<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">a<\/span><span class=\"mbin\">\u2228<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">b<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">1<\/span><\/span><\/span><\/span><\/p>\n<\/li>\n<li class=\"\" data-start=\"1748\" data-end=\"1769\">\n<p class=\"\" data-start=\"1750\" data-end=\"1769\"><span class=\"katex\"><span class=\"katex-mathml\">a\u2227b=0a \\land b = 0<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">a<\/span><span class=\"mbin\">\u2227<\/span><\/span><span class=\"base\"><span class=\"mord mathnormal\">b<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\">0<\/span><\/span><\/span><\/span><\/p>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<li class=\"\" data-start=\"1771\" data-end=\"1964\">\n<p class=\"\" data-start=\"1774\" data-end=\"1795\"><strong data-start=\"1774\" data-end=\"1794\">Complete Lattice<\/strong>:<\/p>\n<ul data-start=\"1799\" data-end=\"1964\">\n<li class=\"\" data-start=\"1799\" data-end=\"1840\">\n<p class=\"\" data-start=\"1801\" data-end=\"1840\"><span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\">Every subset has both a join and a meet.<\/span><\/p>\n<\/li>\n<li class=\"\" data-start=\"1844\" data-end=\"1964\">\n<p class=\"\" data-start=\"1846\" data-end=\"1964\"><span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\">All finite lattices are complete.<\/span><\/p>\n<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<hr class=\"\" data-start=\"1966\" data-end=\"1969\" \/>\n<h2 class=\"\" data-start=\"1971\" data-end=\"2007\">\ud83e\uddea <strong data-start=\"1977\" data-end=\"2007\">Example: Power Set Lattice<\/strong><\/h2>\n<p class=\"\" data-start=\"2009\" data-end=\"2063\">Consider the set <span class=\"katex\"><span class=\"katex-mathml\">A={1,2}A = \\{1, 2\\}<\/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=\"mpunct\">,<\/span><span class=\"mord\">2<\/span><span class=\"mclose\">}<\/span><\/span><\/span><\/span>. Its power set is:<\/p>\n<p class=\"\" data-start=\"2065\" data-end=\"2143\"><span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\">P(A)={\u2205,{1},{2},{1,2}}\\mathcal{P}(A) = \\{\\emptyset, \\{1\\}, \\{2\\}, \\{1,2\\}\\}<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathcal\">P<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">A<\/span><span class=\"mclose\">)<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mopen\">{<\/span><span class=\"mord\">\u2205<\/span><span class=\"mpunct\">,<\/span><span class=\"mopen\">{<\/span><span class=\"mord\">1<\/span><span class=\"mclose\">}<\/span><span class=\"mpunct\">,<\/span><span class=\"mopen\">{<\/span><span class=\"mord\">2<\/span><span class=\"mclose\">}<\/span><span class=\"mpunct\">,<\/span><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><\/span><\/span><\/p>\n<p class=\"\" data-start=\"2145\" data-end=\"2223\"><span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\">With the subset relation (<span class=\"katex\"><span class=\"katex-mathml\">\u2286\\subseteq<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mrel\">\u2286<\/span><\/span><\/span><\/span>), this forms a lattice where:<\/span><\/p>\n<ul data-start=\"2225\" data-end=\"2310\">\n<li class=\"\" data-start=\"2225\" data-end=\"2263\">\n<p class=\"\" data-start=\"2227\" data-end=\"2263\"><strong data-start=\"2227\" data-end=\"2248\">Join (<span class=\"katex\"><span class=\"katex-mathml\">\u2228\\lor<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">\u2228<\/span><\/span><\/span><\/span>)<\/strong>: Union of sets<\/p>\n<\/li>\n<li class=\"\" data-start=\"2264\" data-end=\"2310\">\n<p class=\"\" data-start=\"2266\" data-end=\"2310\"><strong data-start=\"2266\" data-end=\"2288\">Meet (<span class=\"katex\"><span class=\"katex-mathml\">\u2227\\land<\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">\u2227<\/span><\/span><\/span><\/span>)<\/strong>: Intersection of sets<\/p>\n<\/li>\n<\/ul>\n<p class=\"\" data-start=\"2312\" data-end=\"2430\"><span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\">This lattice is bounded, distributive, complemented, and complete.<\/span><\/p>\n<hr class=\"\" data-start=\"2432\" data-end=\"2435\" \/>\n<h2 class=\"\" data-start=\"2437\" data-end=\"2473\">\ud83c\udfa5 <strong data-start=\"2443\" data-end=\"2473\">Recommended Video Lectures<\/strong><\/h2>\n<p class=\"\" data-start=\"2475\" data-end=\"2576\">For a visual and detailed explanation of lattice properties, you may refer to the following lectures:<\/p>\n<ul data-start=\"2578\" data-end=\"2910\">\n<li class=\"\" data-start=\"2578\" data-end=\"2737\">\n<p class=\"\" data-start=\"2580\" data-end=\"2737\"><strong data-start=\"2580\" data-end=\"2617\">Properties of Lattice with Proofs<\/strong>: <span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\">This lecture discusses various properties of lattices with proofs.<\/span><\/p>\n<\/li>\n<li class=\"\" data-start=\"2739\" data-end=\"2910\">\n<p class=\"\" data-start=\"2741\" data-end=\"2910\"><strong data-start=\"2741\" data-end=\"2790\">Properties of Lattice in Discrete Mathematics<\/strong>: <span class=\"relative -mx-px my-[-0.2rem] rounded px-px py-[0.2rem] transition-colors duration-100 ease-in-out\">This video covers properties of lattices in discrete mathematics.<\/span><\/p>\n<\/li>\n<\/ul>\n<hr class=\"\" data-start=\"2912\" data-end=\"2915\" \/>\n<p class=\"\" data-start=\"2917\" data-end=\"3030\">If you need further examples, practice problems, or explanations on specific types of lattices, feel free to ask!<\/p>\n<h3 data-start=\"2917\" data-end=\"3030\"><a href=\"https:\/\/www2.cs.uh.edu\/~arjun\/courses\/ds\/DiscMaths4CompSc.pdf\" target=\"_blank\" rel=\"noopener\">Day 03Part 15-Discrete mathematics for computer science-Properties of lattices with example.<\/a><\/h3>\n<h3 class=\"LC20lb MBeuO DKV0Md\"><a href=\"https:\/\/mrcet.com\/downloads\/digital_notes\/CSE\/II%20Year\/DISCRETE%20MATHEMATICS%20NOTES.pdf\" target=\"_blank\" rel=\"noopener\">DIGITAL NOTES ON Discrete Mathematics B.TECH II YEAR<\/a><\/h3>\n<h3 class=\"LC20lb MBeuO DKV0Md\"><a href=\"https:\/\/www.cs.yale.edu\/homes\/aspnes\/classes\/202\/notes.pdf\" target=\"_blank\" rel=\"noopener\">Notes on Discrete Mathematics<\/a><\/h3>\n","protected":false},"excerpt":{"rendered":"<p>Day 03Part 15-Discrete mathematics for computer science-Properties of lattices with example. [fvplayer id=&#8221;216&#8243;] Here is a concise overview of lattices in discrete mathematics, focusing on their properties with examples, suitable for GATE CSE\/IT preparation: \ud83d\udcd8 What is a Lattice? A lattice is a partially ordered set (POSET) in which every pair of elements has both: [&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-3013","post","type-post","status-publish","format-standard","hentry","category-discrete-mathematics"],"_links":{"self":[{"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/posts\/3013","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=3013"}],"version-history":[{"count":0,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/posts\/3013\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/media?parent=3013"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/categories?post=3013"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/tags?post=3013"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}