Introduction Graph theory is a fundamental area of discrete mathematics with extensive applications in computer science. Graphs are used to model networks, relationships, and structures in various domains such as social networks, communication systems, data organization, and more. Understanding graph theory is essential for algorithm design, data structures, network analysis, and solving complex computational problems.
In this lecture, we will explore the foundational concepts of graph theory, progressing through a structured sequence to build a comprehensive understanding:
Introduction Understanding proofs is fundamental in mathematics and computer science. Proofs provide a systematic way to verify the truth of statements and the correctness of algorithms. This lecture will introduce various proof techniques, including direct proofs, proof by contradiction, proof by contrapositive, and mathematical induction. We will also explore recursion and how it relates to induction, as well as Big-O notation for analyzing algorithm efficiency.
Table of Contents Introduction to Proofs Common Mathematical Expressions in Proofs Direct Proofs Proofs Involving Sets Proof by Contrapositive Proof by Contradiction Proof by Cases Mathematical Induction Strong Induction Recursion and Induction Recurrence Relations Big-O Notation Conclusion Practice Problems Additional Notes 1.
Introduction Understanding proofs is fundamental in mathematics and computer science. Proofs provide a systematic way to verify the truth of statements and the correctness of algorithms. This lecture will introduce various proof techniques, including direct proofs, proof by contradiction, proof by contrapositive, and mathematical induction. We will also explore recursion and how it relates to induction, as well as Big-O notation for analyzing algorithm efficiency.
Table of Contents Introduction to Proofs Common Mathematical Expressions in Proofs Direct Proofs Proofs Involving Sets Proof by Contrapositive Proof by Contradiction Proof by Cases Mathematical Induction Strong Induction Recursion and Induction Recurrence Relations Big-O Notation Conclusion Practice Problems Additional Notes 1.
Introduction Logic is the foundation of mathematical reasoning and critical thinking, especially in computer science. It provides a formal system for reasoning about propositions, their validity, and relationships. Understanding logic is essential for algorithm design, programming, and problem-solving in computer science.
This lecture will cover the fundamental principles of propositional logic and predicate logic, progressing from basic to advanced topics:
Propositions Logical Connectives Truth Tables Logical Equivalences Conditional Statements Predicates and Quantifiers Negation of Quantified Statements Nested Quantifiers Free and Bound Variables Translating English to Logical Expressions Expressing Mathematical Statements in Logic Law of Excluded Middle and Proof Techniques 1.
Introduction Probability is a fundamental concept in mathematics that deals with uncertainty and the likelihood of events occurring. It provides a framework for quantifying and reasoning about uncertainty, which is essential in fields like statistics, computer science, finance, engineering, and more.
This comprehensive lecture will cover the foundational principles of probability, progressing from basic to advanced topics:
Introduction to Probability Basic Probability Definitions Rules of Probability Conditional Probability Independence Bayes’ Theorem Random Variables Expectation (Expected Value) Bernoulli Trials and Binomial Distribution Advanced Applications 1.
Introduction Counting is a fundamental aspect of combinatorics and discrete mathematics. It involves determining the number of ways certain events can occur, which is essential in fields like probability, statistics, computer science, and more. This comprehensive lecture will cover the foundational counting principles, progressing from basic to advanced topics:
Sum Rule Product Rule Permutations Combinations Arrangements with Non-Distinct Objects Counting Integer Solutions (Stars and Bars Method) Bijection Principle $k$-to-1 Rule 1.
These questions were collected through Sli.do during the first five lectures of my Fall 2024 semester.
I have used these questions to improve my lecture notes, in terms of chains of dependency and clarity of notations, and continue to use them to refine them.
Theme 1: Notation and Symbols Use of Symbols: When do we use “=” versus the triple equals sign (≡) to indicate equivalence? Can we use the vertical bar “|” to represent “such that” in set-builder notation?
