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8.3 Representing Relations

8.3 Representing Relations. Rosen 6 th ed., Ch. 8. §8.3: Representing Relations. Some ways to represent n -ary relations: With an explicit list or table of its tuples. With a function from the domain to { T , F } . Or with an algorithm for computing this function.

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8.3 Representing Relations

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  1. 8.3 Representing Relations Rosen 6th ed., Ch. 8

  2. §8.3: Representing Relations • Some ways to represent n-ary relations: • With an explicit list or table of its tuples. • With a function from the domain to {T,F}. • Or with an algorithm for computing this function. • Some special ways to represent binary relations: • With a zero-one matrix. • With a directed graph.

  3. 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:

  4. Properties of Relations • Reflexivity: A relation R on A x A is reflexive if for all aA, (a,a) R. • Symmetry: A relation R on AxA is symmetric if (x,y)  R implies (y,x)  R. • Anti-symmetry: A relation on A x A is anti-symmetric if (a,b)  R implies (b,a)  R. Or a = b.

  5. Zero-One Reflexive, Symmetric • Terms: Reflexive, symmetric, and antisymmetric. • These relation characteristics are very easy to recognize by inspection of the zero-one matrix. any-thing anything anything any-thing Reflexive:all 1’s on diagonal Symmetric:all identicalacross diagonal Antisymmetric:all 1’s are acrossfrom 0’s

  6. 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,links). Visually represented using dots for nodes, and arrows for edges. Notice that 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)

  7. Digraph Reflexive, Symmetric It is extremely 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 Antisymmetric:No link isbidirectional These are asymmetric & non-antisymmetric These are non-reflexive & non-irreflexive

  8. Properties of Relations • Transitivity: A relation on A x A is transitive if (a,b)  R and (b,c)  R imply (a,c)  R. Graph??

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