Lecture_15 - MA1100 Lecture 15 Relations Congruence classes...

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1 MA1100 Lecture 15 Relations Congruence classes Integers modulo n Modular arithmetic
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MA1100 Lecture 15 2 Announcement ± Problem set 6 (available) ² For tutorial class next week ² Homework due the week after ± Problem set 5 solutions (available today) ± Mid-term test ² Some comment on test (end of this lecture) ² Solutions of test will be put up in workbin ² Test scripts to be returned next week
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MA1100 Lecture 15 3 Lesson Plan (week 8 to 13) HW4 Tut 7 Functions 19. Number Th. 18. Functions Week 10 Tut 6 Relations 17. Functions 16. Functions Week 9 HW3 Tut 5 Math. Ind. 15. Relations 14. Relations Week 8 Tut 10 Misc Tut 9 Number Th. Tut 8 Functions Tutorial HW5 23. Revision lecture 22. Revision lecture Week 12 Week 13 21. Number Th. 20. Number Th. Week 11 HW Fri Lecture Tue Lecture
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MA1100 Lecture 15 4 Partitions and Equivalence Relation Given an equivalence relation R on a set A, we get a partition of A: A = {a, b, c, d} partition C = { {a, b} , {c} , {d} } R = { (a,a), (b,b), (a,b), (b,a), (c,c), (d,d) } Equivalence relation Equivalence classes: [c] = {c} [d] = {d} We get [a] = [b] = {a, b}
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MA1100 Lecture 15 5 Partitions and Equivalence Relation Given a partition of a set A, we get an equivalence relation R on A : A = {a, b, c, d} Partition C = { {a} , {b, c, d} } R = { equivalence relation We get x ~ y means there exists S œ C such that x œ S and y œ S. (a,a), (b,b), (c,c), (d,d), (b,c), (c,b), (b,d), (d,b), (c,d), (d,c) } i.e. x, y belong to same group in the partition
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MA1100 Lecture 15 6 Partitions and Equivalence Relation Question: { {a} , {b} , {c} } How many different equivalence relations on the set A = {a, b, c} are there? Same as the number of ways to partition A. { {a} , {b , c} } { {b} , {a , c} } { {c} , {a , b} } { {a , b , c} } 5 different partitions Ans: 5
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MA1100 Lecture 15 7 Why Equivalence Relation? Infinite set Some relevant equivalence relation Not easy to work with infinite elements Finite number of equivalence classes Easier to work with finite classes [a] [b] [c] [d] [e]
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Lecture 15 8 Congruence Classes Definition Congruence modulo 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. Let n be a positive integer. equivalence class: [n]
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Lecture_15 - MA1100 Lecture 15 Relations Congruence classes...

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