Problem 1 Here are three facts about modular congruence which will become important for us later in the course. 1. Symmetry: For any a, b, n = Z, if a = b (mod n), then b = a (mod n). 2. Reflexivity: For any a, n E Z, it is the case that a = a (mod n). 3. Transitivity: For any a, b, c, n = Z, if a = b (mod n) and b = c (mod n), then a = c (mod n). Part A Here are "proofs" of Symmetry and Reflexivity. These proofs contain the right general idea, but they are written very poorly. Symmetry: a, b, n = Z.a-b=cn, b-a = -cn.b= a (mod_n). Reflexivity: a - a= 0, since this is true for any integer. It is = to On. Fix these proofs. For each of the two, go down the checklist in Chapter 5.3 of Book of Proof. Using this checklist, write correct, beautiful proofs that does not violate any of the rules in the list. Part B Write a correct, beautiful proof of Transitivity which does not violate any of the rules in the list.

Problem 1 Here are three facts about modular congruence which will become important for us later in the course. 1. Symmetry: For any a, b, n = Z, if a = b (mod n), then b = a (mod n). 2. Reflexivity: For any a, n E Z, it is the case that a = a (mod n). 3. Transitivity: For any a, b, c, n = Z, if a = b (mod n) and b = c (mod n), then a = c (mod n). Part A Here are "proofs" of Symmetry and Reflexivity. These proofs contain the right general idea, but they are written very poorly. Symmetry: a, b, n = Z.a-b=cn, b-a = -cn.b= a (mod_n). Reflexivity: a - a= 0, since this is true for any integer. It is = to On. Fix these proofs. For each of the two, go down the checklist in Chapter 5.3 of Book of Proof. Using this checklist, write correct, beautiful proofs that does not violate any of the rules in the list. Part B Write a correct, beautiful proof of Transitivity which does not violate any of the rules in the list.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter2: Systems Of Linear Equations

Section2.4: Applications

Problem 33EQ

See similar textbooks