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Representing Relations - Matrix and Graph Representations

Learn about different ways to represent relations, including using zero-one matrices and directed graphs. Understand the properties of reflexive, symmetric, and antisymmetric relations.

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Representing Relations - Matrix and Graph Representations

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  1. §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 TextDiscrete Mathematics & Its Applications(5th Edition)by Kenneth H. Rosen

  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.

  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!

  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.

  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 representationof the relationLikes:Boys×Girlsrelation is:

  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

  7. 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.

  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 identicalacross diagonal Asymmetric:all 1’s are acrossfrom 0’s

  9. Using Directed Graphs • A directed graph or digraphG=(VG,EG) is a set VGof 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) Graph rep. GR: Matrix representation MR: Joe Susan Fred Mary Mark Sally Node set VG(black dots)

  10. Digraph Reflexive, Symmetric It is easy to recognize the reflexive/irreflexive/ symmetric/antisymmetric properties by graph inspection.            Reflexive:Every nodehas a self-loop Irreflexive:No nodelinks to itself Symmetric:Every link isbidirectional Asymmetric:No link isbidirectional These are not symmetric & not asymmetric These are non-reflexive & non-irreflexive

  11. Obvious questions: Given the connection matrix for two relations, how does one find the connection matrix for • The complement? • The symmetric difference?

  12. Example

  13. Question:

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