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Math 412. Worksheet on Congruence and Congruence Classes in Z.
Professor Karen E. Smith
DEFINITION: Fix a non-zero integer N. We say that a, b ∈ Z are congruent modulo N if N |(a − b). We write a ≡ b mod N for “a is congruent to b modulo N .”
DEFINITION: Fix a non-zero integer N. For a ∈ Z, the congruence class of a modulo N is the subset of Z consisting of all integers congruent to a modulo N ; That is, the congruence class of a modulo N is [a]N := {b ∈ Z | b ≡ a mod N }.
Note here that [a]N is the notation for this congruence class— in particular, [a]N stands for a subset of Z, not a number.
A. WARM-UP: True or False. Justify.
(1) T or F: 5 ≡ 19 mod 7 (2) T or F: − 5 ≡ 20 mod 10, (3) T or F: − 11 ≡ − 26 mod 5, (4) T or F: Any two odd integers are congruent modulo 2. (5) T or F: Any two odd integers are congruent modulo 3.
B. EASY PROOFS:
(1) Show that Congruence Modulo N is an Equivalence Relation. That is, show that (a) a ≡ a mod N (congruence is reflexive); (b) If a ≡ b mod N , then b ≡ a mod N (congruence is symmetric); (c) If a ≡ b mod N and b ≡ c mod N , then a ≡ c mod N (congruence is transitive). (2) Prove that every a ∈ Z is congruent mod N to some r ∈ Z such that 0 ≤ r < N. Hint.^1
C. CONGRUENCE CLASS BASICS:
(1) Given two congruence classes, [a]N and [b]N , prove^2 that as sets EITHER [a]N = [b]N OR [a]N ∩ [b]N = ∅. (2) Explain why there are exactly N non-overlapping equivalence classes modulo N. (3) What are the common English words for the two congruence classes modulo 2? (4) List out the congruence classes modulo 4 without using the symbols 0 , 1 , 2 , or 3. Do it in two ways: one using the notation [a]N and one using set-builder notation.
D. TRUE OR FALSE? JUSTIFY.
(1) 47 ∈ [17] 5. (2) [17] 7 ∩ [23] 7 = ∅. (3) [17] 6 ∩ [19] 7 = ∅. (4) For all integers a, [a] 60 ⊂ [a] 10.
(^1) Division algorithm! (^2) Hint: One form of the contrapositive statement is: if [a]N ∩ [b]N 6 = ∅, then [a]N = [b]N. There are standard
techniques you know from 217 to show two sets are the same.
E. ADDING & MULTIPLYING CONGRUENCE CLASSES. Fix N 6 = 0. Let a, b, c, d ∈ Z.
(1) Show that if a ≡ c mod N and b ≡ d mod N , then (a + b) ≡ (c + d) mod N. (2) Show that if a ≡ c mod N and b ≡ d mod N , then (ab) ≡ (cd) mod N.^3 (3) Discuss with your workmates how to use (1) and (2) to define a natural addition and multiplication on the set of congruence classes modulo N. This is delicate: we want to add/multiply two sets (namely, congruence classes) together to produce a third set. If you make some choices, how do you know that your operations are well-defined? (4) There are exactly two congruence classes mod 2: the set of even numbers and the set of odd numbers. Make addition and multiplication tables for the operations you came up in (3) on the set {even, odd} of all congruence classes mod 2. Is there an additive identity? Is there a multiplicative identity? (5) Compute ([7] 5 + [−9] 5 ). Compute [11] 3 × [−66] 3.
F. Let Zn denote the set of congruence classes modulo n.
(1) How many elements are in the set Zn? What type of mathematical objects are these elements? (2) Let [0] 4 , [1] 4 , [2] 4 , [3] 4 be the elements of Z 4. Write down the addition and multiplication tables for this set (using the operations defined in E). Discuss with your workmates how the following properties can be “seen” in your tables: the commutative property of addition, the commutative property of addition, the existence of a multiplicative identity, the existence of an additive identity. Does every element of Z 4 have an additive inverse? Which elements of Z 4 have a multiplicative inverse.
BONUS: What property of an integer n ensures that every element in Zn (except the congruence class which contains zero) has a a multiplicative inverse? What does multiplicative inverse even mean here?
(^3) Try adding and subtracting a convenient quantity from ab − cd.
