DM unit 3 notes

https://drive.google.com/file/d/1Ktah2lhiuxO4htElmrpeq37r0w5Ot-l7/view?usp=sharing

The document provides an introduction to fundamental concepts in Discrete Mathematics, primarily focusing on Algebraic Structures and Lattices. An Algebraic System consists of a non-empty set and one or more $n$-ary operations defined on that set. Key algebraic structures are defined sequentially by satisfying an increasing number of properties: a Semi-group requires closure and associativity, a Monoid adds an identity element, a Group further includes the inverse property, and an Abelian Group adds commutativity. The document also covers Lattices, which are defined as partially ordered sets (posets) where every pair of elements has a Greatest Lower Bound (GLB) and a Least Upper Bound (LUB). The concept of a Partial Order is defined by the relations being reflexive, anti-symmetric, and transitive. Finally, a brief mention of Boolean Algebra as a lattice with complementation is included.

Here are 5 key topics covered:

• Algebraic Systems: A system consisting of a non-empty set and one or more $n$-ary operations defined on the set, denoted by $\langle S, f_1, f_2 \dots f_n \rangle$.
• Semi Group: An algebraic system $\langle S, * \rangle$ where $S$ is a non-empty set and $*$ is a binary operation, which satisfies the closure and associative properties.
• Monoid: An algebraic system $\langle S, * \rangle$ that satisfies closure, associative, and identity properties.
• Group: An algebraic system $\langle S, * \rangle$ that satisfies closure, associative, identity, and inverse properties.
• Lattices: A partially ordered set (poset) $(L, \le)$ in which every pair of elements has a Greatest Lower Bound (GLB) and a Least Upper Bound (LUB).