lecture16 - MA1100 Lecture 16 Relations Congruence classes...

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MA1100 Lecture 16 Relations Congruence classes Integers modulo n odular arithmetic Modular arithmetic 1 Chartrand: 8.5, 8.6
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r Summary Concepts • Domain & range • Ordered pair representation • Reflexive, symmetric, transitive represent relations pecial types of relations (on A only) • Equivalence relation quivalence class special types of relations (on A only) special types of relations (on A only) • Equivalence class • Partition only applies in equivalence relation Lecture 16 2 represent equivalence relations
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rtiti d E i l R l ti Partitions and Equivalence Relation Question: Q How many different equivalence relations on the set A = {a, b, c} are there? Same as the number of ways to partition A. Lecture 16 3
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h E i l R l ti ? Why Equivalence Relation? Infinite set Some relevant equivalence relation [a] [b] [c] [d] [e] Lecture 16 4 Finite number of equivalence classes
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r C l equivalence class: [n] R = { x œ A | (x, n) œ R} Congruence Classes Definition ongruence modulo n lation: Let n be a positive integer. Congruence modulo n relation: 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 5
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r M d l 3 Congruence Modulo 3 onsider the ongruence modulo 3 lation on Consider the congruence modulo 3 relation on Z . [0] 3 = {a œ Z | a ª 0 mod 3} = { - - 0 3 6 9 } { …, 6, 3, 0, 3, 6, 9, … } [1] 3 = {a œ Z | a ª 1 mod 3} = { …, -5, -2, 1, 4, 7, 10, … } [2] = {a œ Z | a ª 2 mod 3} 3 = { …, -4, -1, 2, 5, 8, 11, … } Lecture 16 6 There are exactly 3 distinct congruence classes.
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r M d l Congruence Modulo n Let n be a positive integer. The congruence classes modulo n ] b b mod n} [0] n , [1] n , [2] n , …, [n-1] n [a] n = {b œ Z | b ª a mod n} [n] n , [n+1] n , [n+2] n , …, [2n-1] n [2n] n , [2n+1] n , [2n+2] n , …, [3n-1] n Lecture 16 7 There are exactly n distinct congruence classes.
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r p rti f C r Cl Properties of Congruence Classes Proposition Foreacha ] Let n be a positive integer.
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lecture16 - MA1100 Lecture 16 Relations Congruence classes...

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