1. §7.3 Representing Relations Longin Jan Latecki
11/22/2018
§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.
11/22/2018

2. §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.
11/22/2018

3. 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!
11/22/2018

4. 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.
11/22/2018

5. 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:
11/22/2018

6. 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
11/22/2018

7. • 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.
11/22/2018

8. 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
11/22/2018

9. 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)
11/22/2018

10. 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
11/22/2018

11. 11/22/2018

12. 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?
11/22/2018

13. 11/22/2018

14. Example
11/22/2018

15. Question:
11/22/2018