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4383_Notes_4o2_Zn - Math 4383 Section 4.2 Extra Page 1 of 5...

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Math 4383 Section 4.2 Extra Page 1 of 5 Number Theory Chapter 4 Section 4.2 – Extra Information on Congruence Any set of n integers that are incongruent modulo n is called a complete set of residues mod n. The “standard” set is the set of numbers 0,1,2,…, n-1. For every integer k, define the congruence class of k mod n by ( [ ] { : mod } k x x k n = Remember that congruence is an equivalence relation – which means that [k] = [x] if and only if ( mod k x n THE SETS ARE THE SAME!!!! If we look at all the congruence classes mod n for all the integers, there are only n distinct sets. These can be represented as the congruence classes of 0, 1, 2, 3, … , n-1 Consider the SET where each element is a distinct congruence class mod n. For each distinct set, we use the integer k between 0 and n-1 which is contained in the set as the representative member Call this set “Z n” { } [0],[1],[2],...,[ 1] n n = - Operations corresponding the addition and multiplication are defined by [ ] [ ] [ ] [ ]* [ ] [ ] a b a b a b ab = + = Note “inside the brackets” is simply integer addition and multiplication. Outside the brackets, this is an operation defined with the operands SETS
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