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

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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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t Announcement ± Homework 3 ue today oe o 3 due today ± Online quiz for last week: ² Scores available in Gradebook ² Solutions available in Workbin (Miscellaneous folder) ecture quiz day and Friday ± Lecture quiz today and Friday ± No need to return clickers today est scripts ill be returned to you this ± Test scripts will be returned to you this week during tutorial classes ± Test solutions available in workbin Lecture 16 2 ± Test results
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r Sets A, B Relation R Œ A ä B ~ b b) Summary Concepts a ~ b (a, b) œ R • Ordered pair representation R •Doma in&range dom(R) & range(R) represent relations subsets of A ä B • Reflexive, symmetric, transitive pecial types of relations (on A only) subsets of A & B (resp.) • Equivalence relation quivalence class ] special types of relations (on A only) special types of relations (on A only) • Equivalence class [a] R • Partition only applies in equivalence relation subsets of A Lecture 16 3 represent equivalence relations set of subsets of A
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rtiti d E i l R l ti Partitions and Equivalence Relation Question (from last lecture): Q( ) 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. different partitions ives different equivalence relations { {a} , {b} , {c} } a} b , c} 5 different partitions gives 5 different equivalence relations { {a} , {b , c} } { {b} , {a , c} } c} a , b} Lecture 16 4 { {c} , {a , b} } { {a , b , c} }
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h E i l R l ti ? Why Equivalence Relation? ot easy to Infinite set Not easy to work with infinite Some relevant equivalence relation elements Easier to work with finite lasses [a] [b] [c] [d] [e] Lecture 16 5 Finite number of equivalence classes classes
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r C l equivalence class: [a] R = { x œ A | (x, n) œ R} = { x œ A | (x, a) œ R} Congruence Classes Definition ongruence modulo n lation: Let n > 1 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 6
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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 .
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This note was uploaded on 04/15/2010 for the course MATHS MA1101R taught by Professor Vt during the Spring '10 term at National University of Singapore.

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

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