Equivalence&operation

Equivalence&operation - Equivalence Lecture 6...

Info iconThis preview shows pages 1–14. Sign up to view the full content.

View Full Document Right Arrow Icon
Equivalence Lecture 6 Discrete Mathematical Structures
Background image of page 1

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

View Full DocumentRight Arrow Icon
Equivalence Properties of Relation Reflexivity Symmetry Transitivity Equivalence Equivalence and Partition
Background image of page 2
Reflexivity Relation R on A is Reflexive if for all a A , ( a , a ) R Irreflexive if for all a A, ( a , a ) R Let A ={1,2,3}, R A × A { (1,1) ,(1,3), (2,2) ,(2,1), (3,3) } ? {(1,2),(2,3),(3,1)} ? {(1,2),(2,2),(2,3),(3,1)} ? R is reflexive relation on A if and only if I A R F IR Nothing
Background image of page 3

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

View Full DocumentRight Arrow Icon
Visualized Reflexivity a b c A ={ a,b,c } = 1 1 0 1 1 1 0 0 1 R M
Background image of page 4
Symmetry Relation R on A is Symmetric if whenever ( a , b ) R , then ( b , a ) R Antisymmetric if whenever ( a , b ) R and ( b , a ) R then a = b . Asymmetric if whenever ( a , b ) R then ( b , a ) R (Note: neither anti- nor a-symmetry is the negative of symmetry)
Background image of page 5

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

View Full DocumentRight Arrow Icon
Symmetry Let A ={1,2,3}, R A × A {(1,1),(1,2),(1,3),(2,1),(3,1),(3,3)} symmetric. {(1,2),(2,3),(2,2),(3,1)} antisymmetric. {(1,2),(2,3),(3,1)} antisymmetric and asymmetric. {(11),(2,2)} symmetric and antisymmetric. φ symmetric and antisymmetric, and asymmetric!
Background image of page 6
= 1 1 0 1 0 1 0 1 1 R M Visualized Symmetry A ={ a,b,c } a b c Every edge has its reverse edge m i,j = m j,i
Background image of page 7

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

View Full DocumentRight Arrow Icon
Visualized Antisymmetry A ={ a,b,c } a b c Except cycles, no edge has reverse edge. m i,j + m j,i ≤ 1 for i ≠ j = 1 0 0 1 0 1 0 0 1 R M
Background image of page 8
Visualized Asymmetry A ={ a,b,c } a b c No cycle, no edge has reverse edge. = 0 0 0 1 0 1 0 0 0 R M m i,i = 0 m i,j + m j,i ≤ 1 for i ≠ j
Background image of page 9

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

View Full DocumentRight Arrow Icon
Transitivity Relation R on A is Transitivity if whenever ( a , b ) R, ( b , c ) R, then ( a , c ) R Let A ={1,2,3}, R A × A {(1,1),(1,2),(1,3),(2,1),(2,2),(2,3),(3,3)} is transitive {(1,2),(2,3),(3,1)} is not transitive. {(1,3)} ? φ ? Transitive!
Background image of page 10
Visualized Transitivity a b c A ={ a,b,c } = 1 0 0 1 0 0 1 1 1 R M R is transitive if and only if R n R for all n 1 m i,j =1 and m j,k = 1 then m i,k = 1
Background image of page 11

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

View Full DocumentRight Arrow Icon
What s Wrong? A wrong proof: if R is a symmetric and transitive relation on A , then R must be reflexive. Proof: For any a,b A , if ( a , b ) R , by the symmetry of R , ( b , a ) R ; since R is transitive, ( a , a ) R . So, R is reflexive.
Background image of page 12
Relation R on A is an equivalence relation if and only if it is reflexible, symmetric and transitive. Equality is a special case of equivalence relation. An example: R Z × Z , ( x , y ) R if and only if is integer, i.e. x
Background image of page 13

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

View Full DocumentRight Arrow Icon
Image of page 14
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 03/31/2010 for the course SE C0229 taught by Professor Tao during the Spring '08 term at Nanjing University.

Page1 / 46

Equivalence&operation - Equivalence Lecture 6...

This preview shows document pages 1 - 14. Sign up to view the full document.

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