Day 03Part 10- Discrete mathematics – Introduction of Lattices and and it’s type in Very easy ways.<\/p>\n
[fvplayer id=”221″]<\/p>\n
A Lattice<\/strong> is a type of algebraic structure<\/strong> in Discrete Mathematics that helps in organizing elements in a structured way. It is a partially ordered set (poset)<\/strong> where every two elements<\/strong> have:<\/p>\n Example of a Lattice:<\/strong> Join (\u2228):<\/strong> 2 \u2228 4 = 4<\/strong> (smallest multiple of both) \u00a0If a poset satisfies both Join & Meet properties<\/strong>, it is a Lattice<\/strong>!<\/p>\n A Lattice with the greatest (1) & least (0) elements<\/strong>. A Lattice where Join (\u2228) and Meet (\u2227) distribute over each other<\/strong>: Example: The set {1, 2, 3, 6}<\/strong> under divisibility is distributive.<\/p>\n A Bounded Lattice<\/strong> with a Complement (\u00acA)<\/strong> such that:<\/p>\n \u00a0Example: The Boolean algebra A Lattice that follows a weaker distributive law: \u00a0Example: The subgroup lattice of a group<\/strong> forms a modular lattice.<\/p>\n\n
Consider the set {1, 2, 4, 8}<\/strong> with the divisibility relation (|)<\/strong><\/p>\n\n
Meet (\u2227):<\/strong> 2 \u2227 4 = 2<\/strong> (largest common divisor)<\/p>\n\u00a0Types of Lattices<\/strong><\/h3>\n
\u00a01. Bounded Lattice<\/strong><\/h3>\n
\u00a0Example: {0, 1, 2, 3, 4, 6, 12}<\/code> under divisibility.<\/p>\n\n
\u00a02. Distributive Lattice<\/strong><\/h3>\n
A \u2228 (B \u2227 C) = (A \u2228 B) \u2227 (A \u2228 C)<\/strong>
A \u2227 (B \u2228 C) = (A \u2227 B) \u2228 (A \u2227 C)<\/strong><\/p>\n\u00a03. Complemented Lattice<\/strong><\/h3>\n
\n
{0,1}<\/code> is a Complemented Lattice<\/strong>.<\/p>\n\u00a04. Modular Lattice<\/strong><\/h3>\n
If A \u2264 C, then A \u2228 (B \u2227 C) = (A \u2228 B) \u2227 C<\/strong><\/p>\n\u00a0Key Differences Between Lattice Types<\/strong><\/h3>\n