# lecture15 - This lecture will not be conducted physically...

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1 MA1100 Lecture 15 Relations Equivalence relations Equivalence classes Equivalence relations and partitions Chartrand: 8.3, 8.4 This lecture will not be conducted physically at LT27 on Oct 9. It will only be available on IVLE in webcast format from Oct 9 onward.

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Announcement ± Next week back to normal classroom lecture. ± Homework 3 is due next Tuesday (Oct 13) during lecture. ± Please bring your clickers next Tuesday (Oct 13). All of you are required to return the clickers on that day . ± Return of mid-term test scripts Lecture 15 2
Lecture 15 3 Online quiz ± This lecture comes with an online quiz, which will be available on IVLE from Oct 9, 7am to Oct 11, 11,59pm. ± You may do the online quiz while viewing the webcast lecture, or after the lecture. ± View the webcast for this lecture for more detailed instruction.

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Lecture 15 4 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 . (i) P Q is true (ii) P Q S is true (iii) Conclude S is true
Lecture 15 5 Reflexive, Symmetric, Transitive False Statement Let R be a relation on A. If R is symmetric and transitive , then R is reflexive . Counter-example

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Lecture 15 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 dom(R) = A R is symmetric R is transitive Hypothesis R is reflexive Let a œ A( a , a ) œ R
Lecture 15 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. Relation Set A Refl. Sym. Trans. Equiv. x < y R No No Yes x = y R Yes Yes Yes m | n Z Yes No Yes a ª b mod n Z Yes Yes Yes S Œ T P (U) Yes No Yes

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Lecture 15 8 Parallel lines Define a relation P on set of all lines in the xy- plane : l 1 ~ l 2 if and only if l 1 is parallel to l 2 or l 1 = l 2 . So P is an equivalence relation. reflexive symmetric transitive From previous lecture,
Lecture 15 9 Differ by Integer Define a relation R on Q : a ~ b if and only if a – b œ Z So R is an equivalence relation.

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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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lecture15 - This lecture will not be conducted physically...

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