PPT-22C:19 Discrete Math Integers and Modular Arithmetic

Fall 2011 Sukumar Ghosh Preamble Historically number theory has been a beautiful area of study in pure mathematics However in modern times number theory is very important in the . Sets and Functions. Fall . 2011. Sukumar Ghosh. What is a set?. Definition. . A set is an unordered collection of objects.. . S = {2, 4, 6, 8, …}. . COLOR = {red, blue, green, yellow}. Each object is called an element or a member of the set.. Nikolaj . Bjørner. Microsoft Research. Bit-Precise Constraints: . Applications and . Decision Procedures. Tutorial Contents. Bit-vector decision procedures by categories. Bit-wise operations . Vector Segments. Sequence and Sums. Fall 2011. Sukumar Ghosh. Sequence. A sequence is an . ordered. list of elements. . Examples of Sequence. Examples of Sequence. Examples of Sequence. Not all sequences are arithmetic or geometric sequences.. CS 2800. Prof. Bart Selman. selman@cs.cornell.edu. Module . Basic Structures: Functions and Sequences. . Functions. Suppose we have: . How do you describe the yellow function. ?. What’s a function ?. Nicole Rosenfeld. Different Number systems. O. ne. Two . Two and one. Two two’s. Much. Put up your hands. What would people have done in… . . Rome. . Dubai. . Beijing. . Washington. . Tokyo. Integers and Modular Arithmetic . Fall 2010. Sukumar Ghosh. Preamble. Historically, . number theory . has been a beautiful area of . study in . pure mathematics. . However, in modern times, . number theory is very important in the . Introductory Lecture. What is Discrete Mathematics?. Discrete mathematics is the part of mathematics devoted to the study of discrete (as opposed to continuous) objects.. Calculus deals with continuous objects and is not part of discrete mathematics. . A Conceptual Approach for Teaching Arithmetic & Prealgebra. Barbara Lontz, Assistant Professor of Mathematics. Medea Rambish, Dean of Support Services . Overview. Discuss the national problem of under-preparedness . Factors and Primes. Recursive division algorithm. MA/CSSE 473 . Day 05. Student Questions. One more proof by strong induction. List of review topics I don’t plan to cover in class. Continue Arithmetic Algorithms. Fall 2013. Lecture 11: . Modular arithmetic and applications. announcements. Reading assignment. Modular arithmetic. 4.1-4.3, 7. th. edition. 3.4-3.6, 6. th. edition. review: divisibility. Integers a, b, with a ≠ 0, we say that a . Cryptography is the study of methods for sending secret messages.. It involves . encryption. , in which a message, called . plaintext. , is converted into a form, called . ciphertext. , that may be sent over channels possibly open to view by outside parties. The receiver of the ciphertext uses . Lecture 11. https://abstrusegoose.com/353. Announcements. Lots of folks sounded concerned about English proofs in sections.. THAT’S NORMAL. English proofs aren’t easy the first few times (or the next few times…sometimes not even after a decade…) . 2022. Lecture 11. https://abstrusegoose.com/353. Proof By Cases. Let . . Prime. , . . Odd. . PowerOfTwo. Where . PowerOfTwo. Integer. Prove . We need two different arguments – one for 2 and one for all the other primes…. Factoring: Given a number N , express it as a product of its prime factors. .. . Primality. : Given a number N, determine whether it is a prime. . Factoring is hard. Despite centuries of efforts the fastest methods for factoring a number N take time exponential in number of bits of N..