§7.3 Representing Relations Longin Jan Latecki 6/30/2019 §7.3 Representing Relations Longin Jan Latecki Slides adapted from Kees van Deemter who adopted them from Michael P. Frank’s Course Based on the Text Discrete Mathematics & Its Applications (5th Edition) by Kenneth H. Rosen A word about organization: Since different courses have different lengths of lecture periods, and different instructors go at different paces, rather than dividing the material up into fixed-length lectures, we will divide it up into “modules” which correspond to major topic areas and will generally take 1-3 lectures to cover. Within modules, we have smaller “topics”. Within topics are individual slides. The instructor can bring several modules to each lecture with him, to make sure he has enough material to fill the lecture, or in case he wants to preview or review slides from upcoming or recent past lectures. 6/30/2019

§7.3: Representing Relations Some ways to represent n-ary relations: With a list of tuples. With a function from the domain to {T,F}. Special ways to represent binary relations: With a zero-one matrix. With a directed graph. 6/30/2019

Why bother with alternative representations? Is one not enough? One reason: some things are easier using one representation, some things are easier using another It’s often worth playing around with different representations! 6/30/2019

Connection or (Zero-One) Matrices Let R be a relation from A = {a1, a2, . . . , am} to B = {b1, b2, . . . , bn}. Definition: An m x n connection matrix M for R is defined by Mij = 1 if <ai, bj> is in R, Mij = 0 otherwise. 6/30/2019

Using Zero-One Matrices To represent a binary relation R:A×B by an |A|×|B| 0-1 matrix MR = [mij], let mij = 1 iff (ai,bj)R. E.g., Suppose Joe likes Susan and Mary, Fred likes Mary, and Mark likes Sally. Then the 0-1 matrix representation of the relation Likes:Boys×Girls relation is: 6/30/2019

Special case 1-0 matrices for a relation on A (that is, R:A×A) Convention: rows and columns list elements in the same order 6/30/2019

• R is reflexive iff Mii = 1 for all i. Theorem: Let R be a binary relation on a set A and let M be its connection matrix. Then • R is reflexive iff Mii = 1 for all i. • R is symmetric iff M is a symmetric matrix: M = MT • R is antisymetric if Mij = 0 or Mji = 0 for all i ≠ j. 6/30/2019

Zero-One Reflexive, Symmetric Recall: Reflexive, irreflexive, symmetric, and asymmetric relations. These relation characteristics are very easy to recognize by inspection of the zero-one matrix. any- thing any- thing anything anything any- thing any- thing Reflexive: only 1’s on diagonal Irreflexive: only 0’s on diagonal Symmetric: all identical across diagonal Asymmetric: all 1’s are across from 0’s 6/30/2019

Using Directed Graphs A directed graph or digraph G=(VG,EG) is a set VG of vertices (nodes) with a set EGVG×VG of edges (arcs). Visually represented using dots for nodes, and arrows for edges. A relation R:A×B can be represented as a graph GR=(VG=AB, EG=R). Edge set EG (blue arrows) Matrix representation MR: Graph rep. GR: Joe Susan Fred Mary Mark Sally Node set VG (black dots) 6/30/2019

Digraph Reflexive, Symmetric It is easy to recognize the reflexive/irreflexive/ symmetric/antisymmetric properties by graph inspection.            Reflexive: Every node has a self-loop Irreflexive: No node links to itself Symmetric: Every link is bidirectional Asymmetric: No link is bidirectional These are not symmetric & not asymmetric These are non-reflexive & non-irreflexive 6/30/2019

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Given the connection matrix for two relations, how does Obvious questions: Given the connection matrix for two relations, how does one find the connection matrix for • The complement? • The symmetric difference? 6/30/2019

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Example 6/30/2019

Question: 6/30/2019