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What is Mathematics, really?

Explore the fascinating world of mathematics, where concepts transcend numbers and delve into symbols, objects, images, and more. This course covers discrete structures, mathematical reasoning, combinatorial analysis, algorithmic thinking, and application and modeling. Learn the language of rational thought and its wide range of applications in computer science and other fields.

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What is Mathematics, really?

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  1. What is Mathematics, really? • It’s not just about numbers! • Mathematics is much more than that: • But, these concepts can be about numbers, symbols, objects, images, sounds, anything! Mathematics is, most generally, the study of any and allabsolutely certain truths about any and allperfectly well-defined concepts. Yung-Ling Lai

  2. So, what’s this class about? What are “discrete structures” anyway? • “Discrete” ( “discreet”!) - Composed of distinct, separable parts. (Opposite of continuous.) • “Structures” - Objects built up from simpler objects according to some definite pattern. • “Discrete Mathematics” - The study of discrete, mathematical objects and structures. Yung-Ling Lai

  3. Discrete Mathematics • Mathematical reasoning: think logically; know how to prove • Combinatorial analysis: know how to count • Discrete structures: represent object and their relationships • Algorithmic thinking: how to solve problems by a compute • Application and modeling: model application and solve relevant problems Yung-Ling Lai

  4. Propositions Predicates Proofs Sets Functions Orders of Growth Algorithms Integers Summations Sequences Strings Permutations Combinations Relations Graphs Trees Logic Circuits Automata Discrete Structures We’ll Study Yung-Ling Lai

  5. Some Notations We’ll Learn Yung-Ling Lai

  6. Why Study Discrete Math? • The basis of all of digital information processing is: Discrete manipulations of discrete structures represented in memory. • It’s the basic language and conceptual foundation for all of computer science. • Discrete math concepts are also widely used throughout math, science, engineering, economics, biology, etc., … • A generally useful tool for rational thought! Yung-Ling Lai

  7. Advanced algorithms & data structures Programming language compilers & interpreters. Computer networks Operating systems Computer architecture Database management systems Cryptography Error correction codes Graphics & animation algorithms, game engines, etc.… I.e., the whole field! Uses for Discrete Math in Computer Science Yung-Ling Lai

  8. Course Objectives • Upon completion of this course, the student should be able to: • Check validity of simple logical arguments (proofs). • Check the correctness of simple algorithms. • Creatively construct simple instances of valid logical arguments and correct algorithms. • Describe the definitions and properties of a variety of specific types of discrete structures. • Correctly read, represent and analyze various types of discrete structures using standard notations. Think! Yung-Ling Lai

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