Download presentation

Presentation is loading. Please wait.

Published bySofia Coulton Modified over 3 years ago

1
Representing Relations Rosen 7.3

2
Using Matrices For finite sets we can use zero-one matrices. Elements of each set A and B must be listed in some particular (but arbitrary) order. When A=B we use the same ordering for A and B. m ij = 1 if (a i,b j ) R = 0 if (a i,b j ) R

3
Example Zero-One Matrix b1b2b3 a1 a2 a3 R = {(a1,b1), (a1,b2), (a2,b2), (a3,b2), (a3,b3)}

4
Matrix of a relation on a set, A Can be used to determine whether the relations has certain properties. Recall that R on A is reflexive if (a,a) R for every element a A. ReflexiveNot Reflexive

5
A relation R on a set A is called Symmetric if (b,a) R whenever (a,b) R for a,b A. M R = (M R ) t is Antisymmetric if (a,b) R and (b,a) R only if a=b for a,b A is antisymmetric. –If m ij = 1, i j, m ji = 0 SymmetricAntisymmetricNeither

6
Examples Reflexive Symmetric Reflexive Antisymmetric

7
Let R1, R2 be relations on A A = {1,2,3} R1 = {(1,1), (1,3), (2,1), (3,3)} R2 = {(1,1), (1,2), (1,3), (2,2), (2,3), (3,1)}

8
R1 R2, R1 R2 M R1 R2 = M R1 M R2, M R1 R2 = M R1 M R2

9
What is R2 R1? The composite of R 1 and R 2 is the relation consisting of ordered pairs (a,c) where a A, c A, and for which there exists an element b A such that (a,b) R 1 and (b,c) R 2. R2 R1 = {(1,1), (1,2), (1,3), (2,1), (2,2), (2,3), (3,1)} R1 = {(1,1), (1,3), (2,1), (3,3)} R2 = {(1,1), (1,2), (1,3), (2,2), (2,3), (3,1)}

10
Boolean Product Let A = [a ij ] be an m by k zero-one matrix and B = [b ij ] be a k by n zero-one matrix. Then the Boolean Product of A and B denoted by A B is the m by n matrix with i,j entry c ij where c ij = (a i1 b 1j ) (a i2 b 2j ) ... (a ik b kj ).

11
What is R2 R1? R2 R1 = {(1,1), (1,2), (1,3), (2,1), (2,2), (2,3), (3,1)} M R2 R1 =M R1 M R2

12
Directed Graphs (Digraph) A directed graph consists of a set V of vertices together with a set E of ordered pairs of elements of V called edges. –(a,b), a is initial vertex, b is the terminal vertex a b c Reflexive (Loops at all vertices) Symmetric (All edges both ways)

13
Relation R on a set A a b c R = {(a,b), (b,b), (b,c), (c,a), (c,c)} Transitive? No a b c R = {(a,b), (b,b), (b,c), (a,c), (c,c)} Transitive? Yes Rosen, pp. 493-494

14
Relation R on a set A a b c R = {(a,a), (a,c), (b,b), (b,a), (b,c), (c,c)} Reflexive Antisymmetric Transitive

Similar presentations

Presentation is loading. Please wait....

OK

Basic Properties of Relations

Basic Properties of Relations

© 2018 SlidePlayer.com Inc.

All rights reserved.

To ensure the functioning of the site, we use **cookies**. We share information about your activities on the site with our partners and Google partners: social networks and companies engaged in advertising and web analytics. For more information, see the Privacy Policy and Google Privacy & Terms.
Your consent to our cookies if you continue to use this website.

Ads by Google

Ppt on water activity definition Ppt on online banking system in php Ppt on malignant bone tumors Ppt on history of badminton sport Ppt on bluetooth architecture piconets Ppt on punjabi culture in punjabi language What does appt only meanings Ppt on amartya sen Ppt on academic pressure on students Ppt on solar system and stars