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Unformatted text preview: 1 MA1100 Lecture 14 Relations • Equivalence relations • Equivalence classes • Partition MA1100 Lecture 14 2 Announcement ¡ Hand in homework 3 today ¢ Write your name, matric number and tutorial group on the cover page of your homework scripts. ¡ Problem set 6 (available today) ¡ Midterm test ¢ Test result statistics (end of this lecture) ¢ Test scripts to be returned next week MA1100 Lecture 14 3 Lesson Plan (week 7 to 13) No Tut 13. Relations Test Week 7 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 MA1100 Lecture 14 4 Reflexive, Symmetric, Transitive Let ~ be a relation on A. Definition • We say ~ is a reflexive relation on A if ¾ For all x œ A, x ~ x. • We say ~ is a symmetric relation on A if ¾ For all x, y œ A, if x ~ y, then y ~ x. • We say ~ is a transitive relation on A if ¾ For all x, y, z œ A, if x ~ y and y ~ z, then x ~ z. Ordered pair representation = R (x, x) œ R if (x, y) œ R, then (y, x) œ R if (x, y) œ R and (y, z) œ R, then (x, z) œ R Not all relations have these properties The three properties are independent MA1100 Lecture 14 5 Reflexive, Symmetric, Transitive False Statement “Proof” Let R be a relation on A. If R is symmetric and transitive, then R is reflexive. Let x, y œ A. If (x, y) œ R, then (y, x) œ R, since R is symmetric. Now (x, y) œ R and (y, x) œ R imply (x, x) œ R, since R is transitive. Since (x, x) œ R, R is reflexive. P Ø Q is true P ⁄ Q Ø S is true Conclude S is true The “proof” does not show: For all x œ A, (x, x) œ R MA1100 Lecture 14 6 Reflexive, Symmetric, Transitive False Statement Proof  Exercise Let R be a relation on A. If R is symmetric and transitive, then R is reflexive. Suppose dom(R) = A True MA1100 Lecture 14 7 Equivalence Relation Let R be a relation on A. Definition R is said to be an equivalence relation if it is a reflexive, symmetric and transitive relation on A. If R is an equivalence relation on A, and a ~ b, we say that a is equivalent to b (with respect to this relation). MA1100 Lecture 14 8 Equivalence Relation Yes Yes Yes Yes Yes Trans. Yes Yes Yes Yes No Refl. No Yes No Yes No Sym. a ª b mod n congruence modulo n m  n divides x = y equal to x < y less than S Œ T subset of Equiv. Predicate Name No Yes No Yes No MA1100 Lecture 14 9 Equivalence Relation Examples (1) Define a relation P on set of all lines in the xy plane: l 1 P l 2 if and only if l 1 is parallel to l 2 or l 1 = l 2 ....
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This note was uploaded on 03/19/2012 for the course SCIENCE MA1100 taught by Professor Forgot during the Fall '08 term at National University of Singapore.
 Fall '08
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