Lecture_14 - 1 MA1100 Lecture 14 Relations • Equivalence...

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

View Full Document Right Arrow Icon

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

View Full DocumentRight Arrow Icon

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

View Full DocumentRight Arrow Icon

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

View Full DocumentRight Arrow Icon

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

View Full DocumentRight Arrow Icon

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

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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) ¡ Mid-term 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 ....
View Full Document

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.

Page1 / 32

Lecture_14 - 1 MA1100 Lecture 14 Relations • Equivalence...

This preview shows document pages 1 - 10. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online