discrete mathematics assignment 1

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• Prof. Tom Leighton
• Dr. Marten van Dijk

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• Electrical Engineering and Computer Science
• Mathematics

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• Computer Science
• Applied Mathematics
• Discrete Mathematics
• Probability and Statistics

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Mathematics for computer science, course description.

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1.1: An Overview of Discrete Mathematics

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• Page ID 23230

• Harris Kwong
• State University of New York at Fredonia via OpenSUNY

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What is discrete mathematics? Roughly speaking, it is the study of discrete objects. Here, discrete means “containing distinct or unconnected elements.” Examples include:

• Determining whether a mathematical argument is logically correct.
• Studying the relationship between finite sets.
• Counting the number of ways to arrange objects in a certain pattern.
• Analyzing processes that involve a finite number of steps.

Here are a few reasons why we study discrete mathematics:

• To develop our ability to understand and create mathematical arguments.
• To provide the mathematical foundation for advanced mathematics and computer science courses.

In this text, we will cover these six topics:

• Logic and Proof Techniques:  Logic allows us to determine if a certain argument is valid. We will also learn several basic proof techniques.
• Basic Number Theory:  Number theory is one of the oldest branches of mathematics; it studies properties of integers. We will use the proof techniques we learned to prove some basic facts in number theory.
• Sets:  We study the fundamental properties of sets, and again we will use the proof techniques we learned to prove important results in set theory.
• Relations and Functions:  Relations and functions describe the relationship between the elements from two sets. They play a key role in mathematics.
• Combinatorics:  Combinatorics studies the arrangement of objects. For instance, one may ask, in how many ways can we form a five-letter word. It is used in many disciplines beyond mathematics.
• Big O:   A stand-alone topic on growth of functions.

All of these topics are crucial in the development of your mathematical maturity. The importance of some of these concepts may not be apparent at the beginning. As time goes on, you will slowly understand why we cover such topics. In fact, you may not fully appreciate the subjects until you start taking advanced courses in mathematics.

This is a very challenging course partly because of its intensity. We have to cover many topics that appear totally unrelated at first. This is also the first time many students have to study mathematics in depth. You will be asked to write up your mathematical argument clearly, precisely, and rigorously, which is a new experience for most of you.

Learning how to think mathematically is far more important than knowing how to do all the computations. Consequently, the principal objective of this course is to help you develop the analytic skills you need to learn mathematics. To achieve this goal, we will show you the motivation behind the ideas, explain the results, and dissect why some solution methods work while others do not.

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Discrete mathematics Assignment-1

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Given a set of cycles C of a graph G, the tree graph of G defined by C is the graph T(G,C) whose vertices are the spanning trees of G and in which two trees R and S are adjacent if the union of R and S contains exactly one cycle and this cycle lies in C. Li et al [Discrete Math 271 (2003), 303--310] proved that if the graph T(G,C) is connected, then C cyclically spans the cycle space of G. Later, Yumei Hu [Proceedings of the 6th International Conference on Wireless Communications Networking and Mobile Computing (2010), 1--3] proved that if C is an arboreal family of cycles of G which cyclically spans the cycle space of a $2$-connected graph G, then T(G, C) is connected. In this note we present an infinite family of counterexamples to Hu's result.

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In graph theory connectivity is stated, prevailingly, in terms of paths. While exploiting a proof assistant to check formal reasoning about graphs, we chose to work with an alternative characterization of connectivity: for, within the framework of the underlying set theory, it requires virtually no preparatory notions. We say that a graphs devoid of isolated vertices is connected if no subset of its set of edges, other than the empty set and the set of all edges, is vertex disjoint from its complementary set. Before we can work with this notion smoothly, we must prove that every connected graph has a non-cut vertex, i.e., a vertex whose removal does not disrupt connectivity. This paper presents such a proof in accurate formal terms and copes with hypergraphs to achieve greater generality.

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Discrete Mathematics Tutorial

Mathematical logic.

• Propositional Logic
• Discrete Mathematics - Applications of Propositional Logic
• Propositional Equivalences
• Predicates and Quantifiers
• Mathematics | Some theorems on Nested Quantifiers
• Rules of Inference
• Mathematics | Introduction to Proofs

Sets and Relations

• Set Theory - Definition, Types of Sets, Symbols & Examples
• Types Of Sets
• Irreflexive Relation on a Set
• Reflexive Relation on Set
• Transitive Relation on a Set
• Set Operations
• Types of Functions
• Mathematics | Sequence, Series and Summations
• Representation of Relation in Graphs and Matrices
• What are Relations?
• Closure of Relations
• Mathematical Induction
• Pigeonhole Principle
• Mathematics | Generalized PnC Set 1
• Discrete Maths | Generating Functions-Introduction and Prerequisites
• Inclusion Exclusion principle and programming applications

Boolean Algebra

• Properties of Boolean Algebra
• Number of Boolean functions
• Minimization of Boolean Functions
• Linear Programming

Ordered Sets & Lattices

• Elements of POSET
• Partial Order Relation on a Set
• Axiomatic Approach to Probability
• Properties of Probability

Probability Theory

• Mathematics | Probability Distributions Set 1 (Uniform Distribution)
• Mathematics | Probability Distributions Set 2 (Exponential Distribution)
• Mathematics | Probability Distributions Set 3 (Normal Distribution)
• Mathematics | Probability Distributions Set 5 (Poisson Distribution)

Graph Theory

• Graph and its representations
• Mathematics | Graph Theory Basics - Set 1
• Types of Graphs with Examples
• Mathematics | Walks, Trails, Paths, Cycles and Circuits in Graph
• How to find Shortest Paths from Source to all Vertices using Dijkstra's Algorithm
• Prim’s Algorithm for Minimum Spanning Tree (MST)
• Kruskal’s Minimum Spanning Tree (MST) Algorithm
• Check whether a given graph is Bipartite or not
• Eulerian path and circuit for undirected graph

Special Graph

• Introduction to Graph Coloring
• Edge Coloring of a Graph
• Check if a graph is Strongly, Unilaterally or Weakly connected
• Biconnected Components
• Strongly Connected Components

Group Theory

• Homomorphism & Isomorphism of Group
• Group Isomorphisms and Automorphisms
• Algebraic Structure

GATE PYQs and LMN

• Last Minute Notes – Discrete Mathematics
• Discrete Mathematics - GATE CSE Previous Year Questions

Discrete Mathematics MCQ

• Propositional and First Order Logic.
• Set Theory & Algebra
• Combinatorics
• Top MCQs on Graph Theory in Mathematics
• Numerical Methods and Calculus

Discrete Mathematics is a branch of mathematics that is concerned with “discrete” mathematical structures instead of “continuous”. Discrete mathematical structures include objects with distinct values like graphs, integers, logic-based statements, etc. In this tutorial, we have covered all the topics of Discrete Mathematics for computer science like set theory , recurrence relation, group theory, and graph theory.

Recent Articles on Discrete Mathematics!

• Introduction to Propositional Logic
• Applications of Propositional Logic
• Propositional and Predicate Logic
• Normal and Principle Forms
• Nested Quantifiers Theorem
• Introduction to Proofs
• Types of Sets
• Rough Set Theory
• Sequence and Summations
• Representations of Matrices and Graphs in Relations
• Types of Relation
• Closure of Relation and Equivalence Relations
• Basics of Counting
• Pascal’s Identity
• Permutations and Combinations
• Generalized Permutations and Combinations
• Generating Functions
• Inclusion-Exclusion Principle
• Discrete Probability Theory
• Boolean Functions
• Boolean Algebraic Theorem
• Number of Boolean Functions

Optimization

• Linear Programming 
• Graphical Solution For Linear Programming
• Simplex Algorithm

Ordered Sets & Lattices

• Partially Ordered Sets
• Hasse Diagrams
• Basic Concepts of Probability
• Probability Axioms
• Conditional Probability
• Bayes’ Theorem
• Uniform Distribution
• Exponential Distribution
• Normal Distribution
• Poisson Distribution
• Introduction to Graph
• Basic terminology of a Graph
• Types of a Graph
• Walks, Trails, Paths, and Circuits
• Graph Distance components
• Cut-Vertices and Cut-Edges
• Bridge in Graph
• Independent sets
• Shortest Path Algorithms [Dijkstra’s Algorithm]
• Application of Graph Theory
• Graph Traversals[DFS]
• Graph Traversals[BFS]
• Characterizations of Trees
• Prim’s Minimum Spanning Tree 
• Kruskal’s Minimum Spanning Tree
• Huffman Codes 
• Tree Traversals
• Traveling Salesman Problem
• Bipartite Graphs  
• Independent Sets and Covering
• Eulerian graphs
• Eulerian graphs- Fleury’s algorithm
• Eulerian graphs- Chinese-Postman-Problem Hamilton
• Matching- Basics, Perfect, Bipartite
• Approximation Algorithms

Vertex Colorings

• Chromatic Numbers, Greedy Coloring Algorithm
• Edge Coloring
• Vizing Theorem
• Planar Graph- Basics, Planarity Testing
• Directed Graphs- Degree Centrality
• Directed Graphs- Weak Connectivity
• Directed Graphs- Strong Components 
• Directed Graphs- Eulerian, Hamilton Directed Graphs
• Directed Graphs- Tarjans’ Algorithm To Find Strongly Connected Component
• Handshaking in Graph Theorem
• Groups, Subgroups, Semi Groups
• Isomorphism, Homomorphism
• Automorphism
• Rings, Integral domains, Fields

Quick Links

• Last-Minute Notes (LMNs)
• Quizzes on Discrete Mathematics

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