4. Congruence Classes

Transcription

1 4 Congruence Classes Definition (p21) The congruence class mod m of a Z is Example With m = 3 we have Theorem For a b Z Proof p22 = {b Z : b a mod m} [0] 3 = { } [1] 3 = { } [2] 3 = { } (i) a b (mod m) if and only if = (ii) a b (mod m) if and only if = Corollary Two congruence classes modulo m are either equal or disjoint Part (i) of this result shows that a class can be labelled with any element from within it So in the example above with m = 3 we have [0] 3 = [3] 3 = [6] 3 = In general by the Division Theorem every n Z can be written as n = qm+r for some 0 r m 1 so n r mod m and we can use the label [r] m in place of [n] m Further if 0 r 1 < r 2 m 1 then 0 < r 2 r 1 < m 1 and so m (r 2 r 1 ) ie r 2 r 1 (mod m) Hence [r 1 ] m [r 2 ] m = by part (ii) of the Theorem Thus in {[r] m : 0 r m 1} we see each congruence class once and only once Definition We write Z m for the set of congruence classes mod m So we could write Z m = {[r] m : 0 r m 1} Example Z = {[0] } Since we can add and multiply elements of Z we can define addition and multiplication on Z m 1

2 Definition For a b Z define + = [a + b] m and = [a b] m Examples [7] 10 + = [1] 10 = [] 10 and [7] 10 = [6] 10 = Theorem Addition and multiplication on Z m are well-defined Eg If = [a ] m and = [b ] m then + = [a ] m + [b ] m and = [a ] m [b ] m Proof p26 but simply a restatement of the theorem on modular arithmetic If we express the result of addition or multiplication as a class [r] m with label 0 r m 1 we can write all results in a multiplication table (even if the operation is addition!) Examples (Z 4 +) (Z 4 ) + In this table we see something never seen in (Z ) namely that we can multiply two non-zero objects to get zero For example = Here is an examples of a divisor of zero If in a problem we are working throughout with one modulus m we often drop the [] m and write simply r in place of [r] m See section 213 of PJE for a discussion of the map [r] m r If we want to be reminded of the modulus we often write r 1 + m r 2 and r 1 m r 2 These tables both have the properties of being closed This means that the tables can be filled in using elements from Z 4 2

3 Example (i) Because the product of two even integers is even the set { } is closed under multiplication modulo 8 Verification (ii) The set { } is not closed under multiplication modulo 8 For example = which is not in the set????? (iii) The set { } is a closed subset of (Z 10 ) Verification Note that we have no divisors of zero Earlier results concerning the arithmetic of integers can be reinterpreted in terms of congruence classes Theorem Assume a and c are integers with (a m) = 1 Then the equation [x] m = has a solution in Z m This solution is unique Proof [x] m = soluble x Z : ax c mod m x t Z : ax + mt = c gcd (a m) c Since gcd (a m) = 1 we have gcd (a m) c and thus [x] m = is soluble 3

4 But further if (x 0 t 0 ) is a solution of ax+mt = c then the general solution is ( ) m (x t) = x 0 + l gcd (a m) t a 0 l l Z gcd (a m) ( ) Thus all solutions to ax c mod m are x x 0 mod Since we are m gcd(am) assuming gcd (a m) = 1 the solutions are of the form x x 0 mod (m) which is one unique class [x 0 ] PJE p29 gives an alternative proof Definition An element of Z m is an invertible element if there exists [a ] m such that [a ] m We say that [a ] m is the inverse of and write [a ] m = [a] 1 m We write Z m for the set of invertible elements in Z m Example In the last Chapter we found that 3 had inverse modulo 93 Thus [3] 93 Z 93 is invertible with inverse [] 93 ie [3] 1 93 = [] 93 Hence [3] 93 Z 93 Question What does Z m look like? The above theorem with c = 1 gives Theorem is invertible if and only if (a m) = 1 Proof simply a restatement above Theorem So we can write Z m = {[r] m : 1 r m (r m) = 1} Note The set Z m is not discussed in PJE Example (i) Z (Z ) is = { } and the multiplication table for It is easy to read off inverses from a table so [1] 1 = [2] 1 = [3] 1 = and [4] 1 = (ii) Z 8 = { } and the multiplication table for (Z 8 ) is 4

5 This time we see that every element is a self inverse So in some fundamental way the tables for (Z 8 ) and (Z ) are different What of the tables for (Z ) and (Z 4 +) written as and + do they not have the same form? We will come back to this at the end of the course From the tables we see that both (Z 8 ) and (Z ) are closed In fact Note For all m N (Z m ) is a closed subset of (Z m ) Proof Let Z m Then there exist [d] m Z m such that and [d] m But then ( ) ([d] m ) = ( [d] m ) = [1] m since [d] m = since Thus [d] m is an inverse of and so is invertible ie Z m As another observation we see that in (Z 8 ) and (Z ) we have the nice property that in every row and every column every element occurs once and only once This property was not seen in (Z 4 ) nor ({ } 8 ) Again we will come back to this at the end of the course Finally from Chapter 4 we have as part of a Theorem: If gcd (a m) = 1 then ab 1 ab 2 mod m if and only if b 1 b 2 mod m This gives Cancellation Law in Z m For [b 1 ] m [b 2 ] m Z m if [b 1 ] m = [b 2 ] m then [b 1 ] m = [b 2 ] m

6 Appendix 1) Another example of a multiplication table is (Z 8 ) 2) The set { } is a closed subset of (Z 10 ) Verification 3) I have suggested that the tables for (Z ) and (Z 4 +) are the same In fact we can give a bijection f : Z Z 4 by f ( ) = f ( ) = f ( ) = f ( ) = And this has the lovely property (and you should check this) that for any a b { } f ([a] [b] ) = f ([a] ) + f ([b] ) 6