Xirius-DISCRETE3-COS203.pdf
Xirius AI
This document, titled "COS203 DISCRETE STRUCTURES III" by Xirius, serves as a comprehensive educational material for a course in discrete mathematics. It meticulously covers fundamental concepts and advanced topics essential for understanding the discrete nature of mathematical structures relevant to computer science and other fields. The material is structured to provide a deep dive into various discrete structures, starting from basic definitions and progressing to complex properties, operations, and applications.
The document begins with a thorough exploration of relations, defining them, detailing their various properties (reflexive, symmetric, antisymmetric, transitive), and classifying them into equivalence relations and partial order relations. It further elaborates on equivalence classes, partitions, and operations on relations, including their matrix and digraph representations. Following this, it transitions to functions, covering their definitions, types (injective, surjective, bijective), composition, and inverse functions, establishing the foundational tools for mapping and transformation.
A significant portion of the document is dedicated to graphs and trees, which are crucial for modeling relationships and hierarchical structures. It introduces various types of graphs, their terminology, different representation methods (adjacency matrix, incidence matrix, adjacency list), and concepts like graph isomorphism, Eulerian and Hamiltonian paths, planar graphs, and graph coloring. The discussion on trees covers their definitions, properties, rooted and binary trees, and algorithms for finding spanning trees (Kruskal's and Prim's). Finally, the document delves into algebraic structures, defining binary operations and exploring semigroups, monoids, groups, subgroups, cosets, Lagrange's Theorem, homomorphisms, isomorphisms, rings, and fields, providing a robust foundation in abstract algebra.
MAIN TOPICS AND CONCEPTS
Relations are fundamental in discrete mathematics, defining connections between elements of sets.
- Definition: A relation $R$ from set $A$ to set $B$ is a subset of the Cartesian product $A \times B$. If $A=B$, it's a relation on $A$. We write $aRb$ if $(a,b) \in R$.
- Properties of Relations on a Set $A$:
* Reflexive: For all $a \in A$, $(a,a) \in R$. (Every element is related to itself).
* Irreflexive: For all $a \in A$, $(a,a) \notin R$. (No element is related to itself).
* Symmetric: For all $a,b \in A$, if $(a,b) \in R$, then $(b,a) \in R$. (If $a$ is related to $b$, then $b$ is related to $a$).
* Antisymmetric: For all $a,b \in A$, if $(a,b) \in R$ and $(b,a) \in R$, then $a=b$. (If $a$ is related to $b$ and $b$ is related to $a$, they must be the same element).
* Transitive: For all $a,b,c \in A$, if $(a,b) \in R$ and $(b,c) \in R$, then $(a,c) \in R$. (If $a$ is related to $b$ and $b$ to $c$, then $a$ is related to $c$).
- Types of Relations:
* Equivalence Relation: A relation that is reflexive, symmetric, and transitive.
* Equivalence Class: For an equivalence relation $R$ on $A$ and an element $a \in A$, the equivalence class of $a$, denoted $[a]_R$ or $[a]$, is the set of all elements $x \in A$ such that $xRa$. That is, $[a] = \{x \in A \mid (x,a) \in R\}$.
* Partition: A collection of non-empty, disjoint subsets of $A$ whose union is $A$. Equivalence classes form a partition of $A$.
* Partial Order Relation (Poset): A relation that is reflexive, antisymmetric, and transitive. A set $A$ with a partial order relation $R$ is called a partially ordered set or poset, denoted $(A, R)$.
- Operations on Relations:
* Union ($R_1 \cup R_2$): $(a,b) \in R_1 \cup R_2$ if $(a,b) \in R_1$ or $(a,b) \in R_2$.
* Intersection ($R_1 \cap R_2$): $(a,b) \in R_1 \cap R_2$ if $(a,b) \in R_1$ and $(a,b) \in R_2$.
* Complement ($\bar{R}$): $\bar{R} = (A \times B) \setminus R$.
* Inverse ($R^{-1}$): $R^{-1} = \{(b,a) \mid (a,b) \in R\}$.
* Composition ($R_2 \circ R_1$):