MATH 245 CH9

# MATH 245 CH9 - DISCRETE MATHEMATICS W W L CHEN c W W L...

This preview shows pages 1–3. Sign up to view the full content.

DISCRETE MATHEMATICS W W L CHEN c ± W W L Chen, 1991, 2008. This chapter is available free to all individuals, on the understanding that it is not to be used for ﬁnancial gain, and may be downloaded and/or photocopied, with or without permission from the author. However, this document may not be kept on any information storage and retrieval system without permission from the author, unless such system is not accessible to any individuals other than its owners. Chapter 9 GROUPS AND MODULO ARITHMETIC 9.1. Addition Groups of Integers Example 9.1.1. Consider the set Z 5 = { 0 , 1 , 2 , 3 , 4 } , together with addition modulo 5. We have the following addition table: + 0 1 2 3 4 0 0 1 2 3 4 1 1 2 3 4 0 2 2 3 4 0 1 3 3 4 0 1 2 4 4 0 1 2 3 It is easy to see that the following hold: (1) For every x,y Z 5 , we have x + y Z 5 . (2) For every x,y,z Z 5 , we have ( x + y ) + z = x + ( y + z ). (3) For every x Z 5 , we have x + 0 = 0 + x = x . (4) For every x Z 5 , there exists x 0 Z 5 such that x + x 0 = x 0 + x = 0. Definition. A set G , together with a binary operation * , is said to form a group, denoted by ( G, * ), if the following properties are satisﬁed: (G1) (CLOSURE) For every x,y G , we have x * y G . (G2) (ASSOCIATIVITY) For every x,y,z G , we have ( x * y ) * z = x * ( y * z ). (G3) (IDENTITY) There exists e G such that x * e = e * x = x for every x G . (G4) (INVERSE) For every x G , there exists an element x 0 G such that x * x 0 = x 0 * x = e . Chapter 9 : Groups and Modulo Arithmetic page 1 of 5

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Discrete Mathematics c ± W W L Chen, 1991, 2008 Here, we are not interested in studying groups in general. Instead, we shall only concentrate on groups that arise from sets of the form Z k = { 0 , 1 ,...,k - 1 } and their subsets, under addition or multiplication modulo k . It is not diﬃcult to see that for every k N , the set Z k forms a group under addition modulo k . Conditions (G1) and (G2) follow from the corresponding conditions for ordinary addition and results on
This is the end of the preview. Sign up to access the rest of the document.

## This note was uploaded on 12/20/2009 for the course MATH 245 taught by Professor ramaswamy during the Spring '08 term at San Diego State.

### Page1 / 5

MATH 245 CH9 - DISCRETE MATHEMATICS W W L CHEN c W W L...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online