# 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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