Day 03Part 07-Discretse mathematics for cse-Finding of maxiamal, minimal, maximum, minimum points.<\/p>\n
[fvplayer id=”224″]<\/p>\n
In Discrete Mathematics<\/strong>, finding maximal, minimal, maximum, and minimum points is essential in set theory, lattice theory, and optimization. Let’s break down these concepts:<\/p>\n In a partially ordered set (poset)<\/strong> (S,\u2264)(S, \\leq)<\/span>(<\/span>S<\/span>,<\/span>\u2264<\/span><\/span>)<\/span><\/span><\/span><\/span>, elements are compared based on their ordering relation.<\/p>\n An element a\u2208Sa \\in S<\/span>a<\/span>\u2208<\/span><\/span>S<\/span><\/span><\/span><\/span> is maximal<\/strong> if there is no element b\u2208Sb \\in S<\/span>b<\/span>\u2208<\/span><\/span>S<\/span><\/span><\/span><\/span> such that a<ba < b<\/span>a<\/span><<\/span><\/span>b<\/span><\/span><\/span><\/span>. An element a\u2208Sa \\in S<\/span>a<\/span>\u2208<\/span><\/span>S<\/span><\/span><\/span><\/span> is minimal<\/strong> if there is no element b\u2208Sb \\in S<\/span>b<\/span>\u2208<\/span><\/span>S<\/span><\/span><\/span><\/span> such that b<ab < a<\/span>b<\/span><<\/span><\/span>a<\/span><\/span><\/span><\/span>. If a maximal<\/strong> (or minimal) element is the greatest (or least) element of the set<\/strong>, it is called a maximum<\/strong> (or minimum).<\/p>\n An element mm<\/span>m<\/span><\/span><\/span><\/span> is maximum<\/strong> if m\u2265xm \\geq x<\/span>m<\/span>\u2265<\/span><\/span>x<\/span><\/span><\/span><\/span> for all x\u2208Sx \\in S<\/span>x<\/span>\u2208<\/span><\/span>S<\/span><\/span><\/span><\/span>. An element mm<\/span>m<\/span><\/span><\/span><\/span> is minimum<\/strong> if m\u2264xm \\leq x<\/span>m<\/span>\u2264<\/span><\/span>x<\/span><\/span><\/span><\/span> for all x\u2208Sx \\in S<\/span>x<\/span>\u2208<\/span><\/span>S<\/span><\/span><\/span><\/span>. Consider the set S={2,4,6,8,10}S = \\{2, 4, 6, 8, 10\\}<\/span>S<\/span>=<\/span><\/span>{<\/span>2<\/span>1. Maximal and Minimal Elements<\/strong><\/h3>\n
Maximal Element:<\/strong><\/h3>\n
Example:<\/strong> In the set S={1,3,5,7}S = \\{1, 3, 5, 7\\}<\/span>S<\/span>=<\/span><\/span>{<\/span>1<\/span>,<\/span>3<\/span>,<\/span>5<\/span>,<\/span>7<\/span>}<\/span><\/span><\/span><\/span> ordered by divisibility, 7 is maximal<\/strong> (since no element is greater in divisibility order).<\/p>\nMinimal Element:<\/strong><\/h3>\n
Example:<\/strong> In the same set S={1,3,5,7}S = \\{1, 3, 5, 7\\}<\/span>S<\/span>=<\/span><\/span>{<\/span>1<\/span>,<\/span>3<\/span>,<\/span>5<\/span>,<\/span>7<\/span>}<\/span><\/span><\/span><\/span>, 1 is minimal<\/strong> (since no element divides it except itself).<\/p>\n2. Maximum and Minimum Elements<\/strong><\/h3>\n
Maximum Element:<\/strong><\/h3>\n
Example:<\/strong> In S={1,2,3,4,5}S = \\{1, 2, 3, 4, 5\\}<\/span>S<\/span>=<\/span><\/span>{<\/span>1<\/span>,<\/span>2<\/span>,<\/span>3<\/span>,<\/span>4<\/span>,<\/span>5<\/span>}<\/span><\/span><\/span><\/span>, the maximum<\/strong> is 5<\/strong>.<\/p>\nMinimum Element:<\/strong><\/h3>\n
Example:<\/strong> In S={1,2,3,4,5}S = \\{1, 2, 3, 4, 5\\}<\/span>S<\/span>=<\/span><\/span>{<\/span>1<\/span>,<\/span>2<\/span>,<\/span>3<\/span>,<\/span>4<\/span>,<\/span>5<\/span>}<\/span><\/span><\/span><\/span>, the minimum<\/strong> is 1<\/strong>.<\/p>\nKey Differences:<\/strong><\/h3>\n
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\n Concept<\/strong><\/th>\n Definition<\/strong><\/th>\n Uniqueness<\/strong><\/th>\n<\/tr>\n<\/thead>\n\n \n Maximal<\/strong><\/td>\n No element is strictly greater<\/td>\n May not be unique<\/strong><\/td>\n<\/tr>\n \n Minimal<\/strong><\/td>\n No element is strictly smaller<\/td>\n May not be unique<\/strong><\/td>\n<\/tr>\n \n Maximum<\/strong><\/td>\n The greatest element in the set<\/td>\n Unique if exists<\/strong><\/td>\n<\/tr>\n \n Minimum<\/strong><\/td>\n The smallest element in the set<\/td>\n Unique if exists<\/strong><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n 3. Example in a Partially Ordered Set<\/strong><\/h3>\n