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

Info icon This preview shows pages 1–10. Sign up to view the full content.

View Full Document Right Arrow Icon
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.
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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
Image of page 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.
Image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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
Image of page 4
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
Image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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
Image of page 6
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
Image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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,
Image of page 8
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.
Image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 10
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern