Day 03Part 13-Example based on lattice in discrete mathematics with all the basic concept.<\/p>\n
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A lattice<\/strong> is a partially ordered set (poset) (L, \u2264)<\/strong> where every pair of elements has: Graphically, lattices are represented using Hasse diagrams.<\/strong><\/p>\n \u00a0We check LUB (Join) and GLB (Meet)<\/strong> for each pair.<\/p>\n \u2714 Since every pair has a Join ( \u2228 )<\/strong> and Meet ( \u2227 )<\/strong>, this forms a lattice<\/strong>.<\/p>\n Hasse Diagram Representation:<\/strong><\/p>\n
1\ufe0f\u20e3 Least Upper Bound (LUB) \u2192 Join ( \u2228 )<\/strong>
2\ufe0f\u20e3 Greatest Lower Bound (GLB) \u2192 Meet ( \u2227 )<\/strong><\/p>\n\u00a0Example 1: Check if a Poset Forms a Lattice<\/strong><\/h3>\n
Given Set:<\/strong> L={1,2,3,6}L = \\{ 1, 2, 3, 6 \\}<\/span>L<\/span>=<\/span><\/span>{<\/span>1<\/span>,<\/span>2<\/span>,<\/span>3<\/span>,<\/span>6<\/span>}<\/span><\/span><\/span><\/span> under divisibility relation (|).<\/strong><\/h3>\n
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\n \nElements (a, b)<\/th>\n Join (a \u2228 b) = LCM<\/th>\n Meet (a \u2227 b) = GCD<\/th>\n<\/tr>\n<\/thead>\n \n (1,2)<\/td>\n LCM(1,2) = 2<\/td>\n GCD(1,2) = 1<\/td>\n<\/tr>\n \n (2,3)<\/td>\n LCM(2,3) = 6<\/td>\n GCD(2,3) = 1<\/td>\n<\/tr>\n \n (2,6)<\/td>\n LCM(2,6) = 6<\/td>\n GCD(2,6) = 2<\/td>\n<\/tr>\n \n (3,6)<\/td>\n LCM(3,6) = 6<\/td>\n GCD(3,6) = 3<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n