soln.hw3.330 - Solutions For Homework 3 Ch. 2 2. Show that...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

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

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Solutions For Homework 3 Ch. 2 2. Show that the set { 5 , 15 , 25 , 35 } is a group under multiplication modulo 40. What is the identity element of this group? Can you see any relationship between this group and U (8)? Answer. We need to follow the definition of a group given on page 43: Notation. Let us denote the operation given in the question, multiplication modulo 40, with and the usual multiplication of integers a and b with ab or ( a )( b ). So, we have a b = ab mod 40 Step 1. Check that the operation defined in the question is a binary operation: 5 15 25 35 5 25 35 5 15 15 35 25 15 5 25 5 15 25 35 35 15 5 35 25 Calculations above show that the operation defined in the question assigns to each ordered pair of elements of { 5 , 15 , 25 , 35 } an element in { 5 , 15 , 25 , 35 } . Therefore, it is a binary operation. Step 2. Check associativity of the given operation: Since we have a ( bc ) = a ( bc ) for every a , b , c in Z , we get a ( bc ) = ( ab ) c for every a , b , c in { 5 , 15 , 25 , 35 } . Clearly, a ( bc ) = ( ab ) c implies that a ( bc ) mod 40 = ( ab ) c mod 40 . So, we get a ( b c ) = a ( b c ) mod 40 (def. of ) = [( a mod 40)(( b c ) mod 40)] mod 40 (pr. of mod) = [( a mod 40)(( bc mod 40) mod 40)] mod 40 (def. of ) = [( a mod 40)( bc mod 40)] mod 40 (pr. of mod) = a ( bc ) mod 40 (pr. of mod) = ( ab ) c mod 40 (assoc. of Z) = [(( ab ) mod 40)( c mod 40)] mod 40 (pr. of mod) = [(( ab ) mod 40) mod 40)( c mod 40)] mod 40 (pr. of mod) = [(( a b ) mod 40)( c mod 40)] mod 40 (def. of ) = ( a b ) c mod 40 (pr. of mod) = ( a b ) c (def. of ) ....
View Full Document

This note was uploaded on 10/27/2009 for the course MATH 330 taught by Professor Staff during the Spring '08 term at Ill. Chicago.

Page1 / 5

soln.hw3.330 - Solutions For Homework 3 Ch. 2 2. Show that...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online