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Relations (1)

Relations (1). Rosen 6 th ed., ch . 8. Binary Relations. Let A , B be any two sets. A binary relation R from A to B , written (with signature) R : A ↔ B , is a subset of A × B . E.g. , let < : N ↔ N :≡ {( n , m ) | n < m }

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Relations (1)

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  1. Relations (1) Rosen 6thed., ch. 8

  2. Binary Relations • Let A, B be any two sets. • A binary relationR from A to B, written (with signature) R:A↔B, is a subset of A×B. • E.g., let <: N↔N :≡ {(n,m)| n < m} • The notation a Rb or aRb means (a,b)R. • E.g., a <b means (a,b) < • If aRb we may say “a is related to b (by relation R)”, or “a relates to b (under relation R)”. • A binary relation R corresponds to a predicate function PR:A×B→{T,F} defined over the 2 sets A,B; e.g., “eats” :≡ {(a,b)| organism a eats food b}

  3. Complementary Relations • Let R:A↔B be any binary relation. • Then, R:A↔B, the complement of R, is the binary relation defined byR :≡ {(a,b) | (a,b)R} = (A×B) − R Note this is just R if the universe of discourse is U = A×B; thus the name complement. • Note the complement of R is R. Example: < = {(a,b) | (a,b)<} = {(a,b) | ¬a<b} = ≥

  4. Inverse Relations • Any binary relation R:A↔B has an inverse relation R−1:B↔A, defined byR−1 :≡ {(b,a) | (a,b)R}. E.g., <−1 = {(b,a) | a<b} = {(b,a) | b>a} = >. • E.g., if R:People→Foods is defined by aRb  aeatsb, then: bR−1a  bis eaten bya. (Passive voice.)

  5. Relations on a Set • A relation on the set A is a relation from A to A. • In other words, a relation on a set A is a subset of AхA. • Example 4. Let A be the set {1, 2, 3, 4}. Which ordered pairs are in the relation R = {(a, b) | a divides b}? Solution: R = {(1, 1), (1, 2), (1, 3), (1, 4), (2, 2), (2, 4), (3, 3), (4, 4)}

  6. Reflexivity • A relation R on A is reflexive if aA,aRa. • E.g., the relation ≥ :≡ {(a,b) | a≥b} is reflexive. • A relation is irreflexive iff its complementary relation is reflexive. (for every aA, (a, a)  R) • Note “irreflexive” ≠ “notreflexive”! • Example: < is irreflexive. • Note: “likes” between people is not reflexive, but not irreflexive either. (Not everyone likes themselves, but not everyone dislikes themselves either.)

  7. Symmetry & Antisymmetry • A binary relation R on A is symmetric iff R = R−1, that is, if (a,b)R ↔ (b,a)R. i.e, ab((a, b)  R → (b, a)  R)) • E.g., = (equality) is symmetric. < is not. • “is married to” is symmetric, “likes” is not. • A binary relation R is antisymmetric if ab(((a, b)  R  (b, a)  R) → (a = b)) • < is antisymmetric, “likes” is not.

  8. Asymmetry • A relation R is called asymmetric if (a,b)R implies that (b,a)  R. • ‘=’ is antisymmetric, but not asymmetric. • < is antisymmetric and asymmetric. • “likes” is not antisymmetric and asymmetric.

  9. Transitivity • A relation R is transitive iff (for all a,b,c) (a,b)R  (b,c)R→ (a,c)R. • A relation is intransitive if it is not transitive. • Examples: “is an ancestor of” is transitive. • “likes” is intransitive. • “is within 1 mile of” is… ?

  10. Examples of Properties of Relations • Example 7. Relations on {1, 2, 3, 4} • R1 = {(1, 1), (1, 2), (2, 1), (2, 2), (3, 4), (4, 1), (4, 4)}, R2 = {(1, 1), (1, 2), (2, 1)}, R3 = {(1, 1), (1, 2), (1, 4), (2, 1), (2, 2), (3, 3), (4, 1), (4, 4)}, R4 = {(2, 1), (3, 1), (3, 2), (4, 1), (4, 2), (4, 3)}, R5 = {(1, 1), (1, 2), (1, 3), (1, 4), (2, 2), (2, 3), (2, 4), (3, 3), (3, 4), (4, 4)}, R6 = {(3, 4)}.

  11. Examples Cont. • Solution • Reflective : X, X, O, X, O, X • Irreflective : X, X, X, O, X, O • Symmetric : X, O, O, X, X, X • Antisymmetric : X, X, X, O, O, O • Asymmetric : X, X, X, O, X, O • Transitive : X, X, X, O, O, O

  12. Examples Cont. • Example 5. Relations on the set of integers • R1 = {(a, b) | a ≤ b}, R2 = {(a, b) | a > b}, R3 = {(a, b) | a = b or a = -b}, R4 = {(a, b) | a = b}, R5 = {(a, b) | a = b + 1}, R6 = {(a, b) | a + b ≤ 3},

  13. Examples Cont. • Solution • Reflective : O, X, O, O, X, X • Irreflective : X, O, X, X, O, X • Symmetric : X, X, O, O, X, O • Antisymmetric : O, O, X, O, O, X • Asymmetric : X, O, X, X, O, X • Transitive : O, O, O, O, X, X

  14. Composition of Relations • DEFINITION: Let R be a relation from a set A to a set B and S a relation from B to a set C. The composite of R and S is the relation consisting of ordered pairs (a, c), where a A, c  C, and for which there exists an element b  B such that (a, b)  R and (b, c)  S. We denote the composite of R and S by S◦R.

  15. Composition of Relations Cont. • R : relation between A and B • S : relation between B and C • S°R : composition of relations R and S • A relation between A and C • {(x,z)| x Î A, z Î C, and there exists y Î B such that xRy and ySz } B C C A S A R z S°R z y x x u w w

  16. Example R S 1 2 3 4 1 2 3 4 1 2 3 1 2 S°R 1 2 3 1 2

  17. Example Cont. • What is the composite of the relations R and S, where R is the relation from {1, 2, 3} to {1, 2, 3, 4} with R = {(1, 1), (1, 4), (2, 3), (3, 1), (3, 4)} and S is the relation from {1, 2, 3, 4} to {0, 1, 2} with S = {(1, 0), (2, 0), (3, 1), (3, 2), (4, 1)}. Solution: S◦R = {(1, 0), (1, 1), (2, 1), (2, 2), (3, 0), (3, 1)}

  18. Power of Relations • DEFINITION: Let R be a relation on the set A. The powers Rn, n = 1, 2, 3, …, are defined recursively by R1 = R and Rn+1 = Rn◦R. • EXAMPLE: Let R = {(1, 1), (2, 1), (3, 2), (4, 3)}. Find the powers Rn, n = 2, 3, 4, …. • Solution: R2 = R◦R = {(1, 1), (2, 1), (3, 1), (4, 2)}, R3 = R2◦R = {(1, 1), (2, 1), (3, 1), (4, 1)}, R4 = R2◦R = {(1, 1), (2, 1), (3, 1), (4, 1)} => Rn = R3

  19. Power of Relations Cont. • THEOREM: The relation R on a set A is transitive if and only if Rn ÍR for n = 1, 2, 3, …. • EXAMPLE: Let R = {(1, 1), (2, 1), (3, 2), (4, 3)}. Is R transitive? => No (see the previous page). • EXAMPLE: Let R = {(2, 1), (3, 1), (3, 2), (4, 1), (4, 2), (4, 3)}. Is R transitive? • Solution: R2 = ?, R3 = ?, R4 = ? => maybe a tedious task^^

  20. Representing Relations • Some special ways to represent binary relations: • With a zero-one matrix. • With a directed graph.

  21. Using Zero-One Matrices • To represent a relation R by a matrix MR = [mij], let mij = 1 if (ai,bj)R, else 0. • E.g., Joe likes Susan and Mary, Fred likes Mary, and Mark likes Sally. • The 0-1 matrix representationof that “Likes”relation:

  22. Zero-One Reflexive, Symmetric • Terms: Reflexive, non-Reflexive, irreflexive,symmetric, asymmetric, and antisymmetric. • 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 Symmetric:all identicalacross diagonal Antisymmetric:all 1’s are acrossfrom 0’s Reflexive:all 1’s on diagonal Irreflexive:all 0’s on diagonal

  23. Finding Composite Matrix • Let the zero-one matrices for S◦R, R, S be M S◦R, MR, MS. Then we can find the matrix representing the relation S◦R by: • M S◦R = MR ⊙ MS • EXAMPLE: MR MS M S◦R ⊙ =

  24. Examples of matrix representation • List the ordered pairs in the relation on {1, 2, 3} corresponding to these matrices (where the rows and columns correspond to the integers listed in increasing order).

  25. Examples Cont. • Determine properties of these relations on {1, 2, 3, 4}. R1 R2 R3

  26. Examples Cont. • Solution • Reflective : X, O, X • Irreflective : O, X, O • Symmetric : O, X, O • Antisymmetric : X, X, X • Asymmetric : X, X, X

  27. Examples Cont. • Transitive : • R12 = ⊙ = • Not Transitive • Notice: (1, 4) and (4, 3) are in R1 but not (1, 3)

  28. Examples Cont. • R22 = ⊙ = • R23 = ⊙ =

  29. Examples Cont. • R24 = ⊙ = • Not transitive • Notice: (1, 3) and (3, 4) are in R1 but not (1, 4)

  30. Examples Cont. • EXAMPLE: Let R = {(2, 1), (3, 1), (3, 2), (4, 1), (4, 2), (4, 3)}. Is R transitive? • Solution: transitive (see below) • R2 ⊙ = • R3 ⊙ = • R33 = ⊙ =

  31. 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) GR MR Joe Susan Fred Mary Mark Sally Node set VG(black dots)

  32. 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 Asymmetric, non-antisymmetric Non-reflexive, non-irreflexive

  33. b b b a a a c c c Example • Determine which are reflexive, irreflexive, symmetric, antisymmetric, and transitive. transitive Reflexive Symmetric Not transitive Reflexive antisymmetric

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