Equivalence Relations - Mathematical and Statistical Sciences
Modular Arithmetic Theorem: For any natural number m, the modular relation ≡ m is an equivalence relation on ℤ. Pf: For any x in ℤ, since x – x = 0 and m | 0, x ≡ m x. (Reflexitivity) If x ≡ m y then m | x – y. Since y – x = -(x-y), m | y – x, and so, y ≡ m x. (Symmetry) If x ≡ m y and y ≡ m z then m | x – y and m | y ...
Download Equivalence Relations - Mathematical and Statistical Sciences
Information
Domain:
Source:
Link to this page:
Documents from same domain
Higher Geometry - Mathematical & Statistical Sciences
www-math.ucdenver.eduThree Point Geometry Axioms for the Three Point Geometry: 1. There exist exactly 3 points in this geometry. 2. Two distinct points are on exactly one line. 3. Not all the points of the geometry …
Geometric Constructions - UC Denver
www-math.ucdenver.eduPhilosophy of Constructions Constructions using compass and straightedge have a long history in Euclidean geometry. Their use reflects the basic axioms of this system.
The Chinese Remainder Theorem - UC Denver
www-math.ucdenver.eduby 3, and remainder 3 when divided by 7. We are looking for a number which satisfies the congruences, x ≡ 2 mod 3, x ≡ 3 mod 7, x ≡ 0 mod 2 and x ≡ 0 mod 5.
Proof Methods - Mathematical and Statistical Sciences
www-math.ucdenver.eduWriting mathematical proofs is therefore an art form (the art of convincing) and a social process since it is directed at people (the readers). A mathematical proof of a statement strongly depends on who the proof is written for. Proofs for a research audience are quite different from those found in textbooks. And even textbook proofs look ...
Linear Codes
www-math.ucdenver.educonvert from the generator matrix to the parity-check matrix and vice-versa. This is easily done. Theorem 2: If G = [I k A] is the generator matrix (in standard form) for the [n,k]-code C, then H = [-AT I n-k] is the parity check matrix for C. Proof.
Related documents
The Euclidean Algorithm and Multiplicative Inverses
www.math.utah.eduthinking about finding multiplicative inverses in modular arithmetic, but it turns out that if you look at his algorithm in reverse, that’s exactly what it does! The Euclidean Algorithm makes repeated used of integer division ideas: We “know” that if a and b are positive integers, then we may write a b = q + r b
Introduction to Shimura Varieties - James Milne
www.jmilne.orgThe arithmetic properties of elliptic modular functions and forms were extensively studied in the 1800s, culminating in the beautiful Kronecker Jugendtraum. Hilbert emphasized the importance of extending this theory to functions of several variables in the twelfth of his famous problems at the International Congress in 1900.
Discrete Structures Lecture Notes - Stanford University
web.stanford.eduDiscrete Structures Lecture Notes Vladlen Koltun1 Winter 2008 1Computer Science Department, 353 Serra Mall, Gates 374, Stanford University, Stanford, CA 94305, USA; vladlen@stanford.edu.
Practice problems for the Math Olympiad - Texas A&M ...
www.tamug.eduModular Arithmetic means recycling of integers when they reach a fixed value, e.g., a 12 hour clock. or integers a, b, n, we write a=b(mod n), read “a is congruent to b modulo n”, if a-b is a multiple of n. e.g., 38=14(mod 12) because 38 -14 = 24 =2*12. Solution: For this question, we can solve it by finding all solutions and proving there ...
The science of encryption: prime numbers and mod arithmetic
math.berkeley.eduThe science of encryption: prime numbers and mod n arithmetic Go check your e-mail. You’ll notice that the webpage address starts with \https://". The \s" at the end stands for \secure" meaning that a process called SSL is being used to encode the contents of your inbox and prevent people from hacking your account. The heart of SSL { as well
Cryptography: An Introduction (3rd Edition)
www.cs.umd.edu(modular arithmetic) and a little probability before. In addition, they would have at some point done (but probably forgotten) elementary calculus. Not that one needs calculus for cryptography, but the ability to happily deal with equations and symbols is certainly helpful. Apart from that I introduce everything needed from scratch.
Abelian Varieties - James Milne
www.jmilne.orgThese notes are an introduction to the theory of abelian varieties, including the arithmetic of abelian varieties and Faltings’s proof of certain finiteness theorems. The orginal version of the notes was distributed during the teaching of an advanced graduate course. Alas, the notes are still in very rough form. BibTeX information @misc{milneAV,
Modular Arithmetic Practice - CMU
www.math.cmu.eduSep 13, 2015 · Modular Arithmetic Practice Joseph Zoller September 13, 2015 Practice Problem Solutions 1. Given that 5x 6 (mod 8), nd x. [Solution: 6] 2. Find the last digit of 7100 [Solution: 1] 7100 (72) 50 49 ( 1)50 1 mod 10. 3. (1992 AHSME 17) The two-digit integers form 19 to 92 are written consecutively to form the large integer N = 192021 909192.