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lecture16(complete)

# lecture16(complete) - MA1100 Lecture 16 Relations...

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MA1100 Lecture 16 Relations Congruence classes Integers modulo n Modular arithmetic 1 Chartrand: 8.5, 8.6

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A t Announcement Homework 3 due today Online quiz for last week: Scores available in Gradebook Solutions available in Workbin (Miscellaneous folder) Lecture quiz today and Friday No need to return clickers today Test scripts will be returned to you this week during tutorial classes Test solutions available in workbin Lecture 16 2 Test results
S r Sets A, B Relation R Œ A ä B a ~ b (a b) œ R Summary Concepts (a, b) • Ordered pair representation R • Domain & range dom(R) & range(R) represent relations subsets of A ä B • Reflexive, symmetric, transitive special types of relations (on A only) subsets of A & B (resp.) • Equivalence relation • Equivalence class [a] special types of relations (on A only) R • Partition only applies in equivalence relation t i l l ti subsets of A t f b t f A Lecture 16 3 represent equivalence relations set of subsets of A

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P rtiti d E i l R l ti Partitions and Equivalence Relation Question (from last lecture): How many different equivalence relations on the set A = {a, b, c} are there? Ans: 5 Same as the number of ways to partition A. 5 different partitions gives 5 different equivalence relations { {a} , {b} , {c} } { {a} , {b , c} } { {b} , {a , c} } { {c} , {a , b} } Lecture 16 4 { {a , b , c} }
Wh E i l R l ti ? Why Equivalence Relation? Not easy to Infinite set work with infinite Some relevant equivalence relation elements Easier to [b] work with finite classes [a] [c] [d] [e] Lecture 16 5 Finite number of equivalence classes

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C r Cl equivalence class: [a] R = { x œ A | (x, n) œ R} = { x œ A | (x, a) œ R} Congruence Classes Definition Congruence modulo n relation: Let n > 1 be a positive integer. a ~ b if and only if a ª b mod n is an equivalence relation on Z . For each a œ Z , we have an equivalence class [a] n = { x œ Z | x ª a mod n}. We call [a] n the congruence class of a modulo n. Lecture 16 6
C r M d l 3 Congruence Modulo 3 Consider the congruence modulo 3 relation on Z . [0] 3 = {a œ Z | a ª 0 mod 3} = { -6 -3 0 3 6 9 } all multiples of 3 = [3] 3 = [6] 3 = [9] 3 = … { …, 6, 3, 0, 3, 6, 9, … } [1] 3 = {a œ Z | a ª 1 mod 3} { 5 2 1 4 7 10 } = [4] 3 = [7] 3 = [10] 3 = … = { …, -5, -2, 1, 4, 7, 10, … } all integers with remainder 1 when divided by 3 [2] 3 = {a œ Z | a ª 2 mod 3} = [5] 3 = [8] 3 = [11] 3 = … = { …, -4, -1, 2, 5, 8, 11, … } all integers with remainder 2 when divided by 3 [5] [8] [11] Lecture 16 7 There are exactly 3 distinct congruence classes.

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C r M d l Congruence Modulo n Let n be a positive integer.
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