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CSCI 115

CSCI 115. Chapter 4 Relations and Digraphs. CSCI 115. §4 .1 Product Sets and Partitions. §4 .1 – Product Sets and Partitions. Product Set Ordered pair Cartesian Product Theorem 4.1.1 For any 2 finite non-empty sets A and B, |A x B| = |A||B|. §4 .1 – Product Sets and Partitions.

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CSCI 115

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  1. CSCI 115 Chapter 4 Relations and Digraphs

  2. CSCI 115 §4.1 Product Sets and Partitions

  3. §4.1 – Product Sets and Partitions • Product Set • Ordered pair • Cartesian Product • Theorem 4.1.1 • For any 2 finite non-empty sets A and B, |A x B| = |A||B|

  4. §4.1 – Product Sets and Partitions • Partitions • A partition (quotient set) of A is a set P of nonempty subsets of A such that: • Each element of A belongs to a set in P • If A1 and A2 are elements of P, then A1A2 = {} • Each element of P is called a cell or a block

  5. CSCI 115 §4.2 Relations and Digraphs

  6. §4.2 – Relations and Digraphs • Relations • Relation from A to B • Subset of A x B • Relation on A • Subset of A x A

  7. §4.2 – Relations and Digraphs • Sets arising from relations • Domains • Domain of relation • Ranges • Range of relation • Range of an element • Range of a subset

  8. §4.2 – Relations and Digraphs • Theorem 4.2.1 • Let R be a relation from A to B, and let A1 and A2 be subsets of A. Then: • If A1 A2, then R(A1)  R(A2) • R(A1 A2) = R(A1)  R(A2) • R(A1 A2)  R(A1)  R(A2) • Theorem 4.2.2 • Let R and S be Relations from A to B. If R(a) = S(a)  a  A, then R = S.

  9. §4.2 – Relations and Digraphs • The matrix of a relation • If A = {a1, a2, …, am} and B = {b1, b2, …, bn}, then a relation R from A to B can be represented by an m x n boolean matrix (MR) constructed as follows: mij = { 1 if (ai, bj)  R0 if (ai, bj)  R

  10. §4.2 – Relations and Digraphs • The Digraph of a relation • Let R be a relation on A. The Digraph of R is constructed as follows: • Draw a circle for each element in A, and label the circles accordingly (these are called vertices) • Draw an arrow from ai to ajiffaiRaj(these are called edges) • In-degrees and out-degrees

  11. §4.2 – Relations and Digraphs • Relation Restriction • Let R be a relation on a set A, with B  A. The restriction of R to B is R  (B x B).

  12. CSCI 115 §4.3 Paths in Relations and Digraphs

  13. §4.3 – Paths in Relations and Digraphs • Path of length n from a to b: • aRx1, x1Rx2, x2Rx3, …, xn-1Rb • Geometric path in digraph • Cycle: A path that begins and ends at the same vertex

  14. §4.3 – Paths in Relations and Digraphs • New relations from paths • Rn: xRny iff there is a path of length n from x to y • R: xRy iff there is any path from x to y • Connectivity Relation • R*: xR*y iff there is a path from x to y, or x = y • Reachability Relation

  15. §4.3 – Paths in Relations and Digraphs • Theorem 4.3.1 • If R is a relation on A = {a1, a2, …, an}, then MR2 = MR MR = ((MR))2 • Theorem 4.3.2 • For n  2 and R a relation on a finite set A, we have MRn = MR MR...  MR (n factors)

  16. §4.3 – Paths in Relations and Digraphs • Path composition •  is used as a variable for paths • Assume we have: • 1: a, x1, x2, …, xn-1, b • 2: b, y1, y2, …, ym-1, c • Then, we can compose 1 and 2 as follows: • 21: a, x1, x2, …, xn-1, b, y1, y2, …, ym-1, c • 21 is a path of length n + m • 12 does not make sense

  17. CSCI 115 §4.4 Properties of Relations

  18. §4.4 – Properties of Relations • A relation R on a set A is: • Reflexive iff (a,a)  Ra  A • Irreflexive iff (a,a)  R a  A • Symmetric iff (a,b)  R  (b,a)  R • Asymmetric iff (a,b)  R  (b,a)  R • Antisymmetric iff (a,b)  R and (b,a)  R  a = b • Transitive iff (a,b)  R and (b,c)  R  (a,c)  R

  19. §4.4 – Properties of Relations • Theorem 4.4.1 • A relation R is transitive iff it satisfies the following property: If there is a path of length greater than 1 from vertex a to vertex b, there is a path of length 1 from a to b (i.e. aRb). In other words, R is transitive iff Rn  R  n > 1.

  20. §4.4 – Properties of Relations • Theorem 4.4.2 • Let R be a relation on a set A. Then: • Reflexivity of R means a  R(a)  a  A • Symmetry of R means a  R(b) iff b  R(a) • Transitivity of R means b  R(a) and c  R(b) implies c  R(a)

  21. §4.4 – Properties of Relations • Other properties: • The digraph of a reflexive relation has a cycle of length 1 at every vertex • The digraph of an irreflexive relation has no cycles of length 1 • If R is reflexive then the Domain(R) = Range(R) = A • The matrix of a reflexive relation has all 1s on the diagonal • The matrix of an irreflexive relation has all 0s on the diagonal • The matrix of a symmetric relation has mij = mjii,j (MR = MRT) • The matrix of an asymmetric relation meets the following 2 criteria: • mij = 1  mji = 0 AND the diagonal must be all 0s • The matrix of an antisymmetric relation meets the following criteria: • If ij, then mij = 0 or mji = 0 • The Matrix of a transitive relation meets the following criteria: • MR must have a 1 everywhere ((MR))2 has a 1

  22. §4.4 – Properties of Relations • Graph of a symmetric relation • Undirected edge • Adjacent vertices • Connected graphs • Disconnected graphs

  23. CSCI 115 §4.5 Equivalence Relations

  24. §4.5 – Equivalence Relations • Equivalence Relation • A relation R on a set A is an equivalence relation if it is reflexive, symmetric, and transitive

  25. §4.5 – Equivalence Relations • Theorem 4.5.1 • Let P be a partition of a set A such that P = {A1, A2, …, An}. Define a relation R on A as follows:aRb iff aAi and bAi (i  {1, 2, …, n}). Then R is an equivalence relation on A. • Here R is called the equivalence relation determined by P.

  26. §4.5 – Equivalence Relations • Lemma 4.5.1 • Let R be an equivalence relation on A. Let a, b  A. Then aRb iff R(a) = R(b). • Theorem 4.5.2 • Let R be an equivalence relation on A, and let P be the collection of all distinct relative sets R(a) for a in A. Then P is a partition of A, and R is the equivalence relation determined by P. • Note: When R is an equivalence relation on A, the sets R(a) are called equivalence classes. The partition constructed in Theorem 4.5.2 is denoted A/R.

  27. §4.5 – Equivalence Relations Procedure to determine A/R (A finite or countable): • Choose any aA and find R(a). • If R(a)  A, choose bA, bR(a) and find R(b). • If A is not the union of the computed equivalence classes, choose xA such that x is not in any of the equivalence classes, and find R(x). • Repeat step 3 until A is accounted for, or a pattern emerges which describes the equivalence classes. • A/R is the partition formed by using each equivalence class as a cell.

  28. CSCI 115 §4.6 Data Structures for Relations and Digraphs

  29. §4.6 – Data Structures for Relations and Digraphs • Linked list data structure • Storage cell • Data • Pointer • Implementations • Structures • Arrays • We will use the array implementation

  30. §4.6 – Data Structures for Relations and Digraphs • Implementation 1 – Start, Tail, Head, Next • Fill in tail and head arrays • Fill in start variable • Fill in next array • The tail, head and next arrays have as many elements as the digraph has edges • Start is a simple variable (not an array)

  31. §4.6 – Data Structures for Relations and Digraphs • Implementation 2 – Vert, Tail, Head, Next • Fill in tail and head arrays • Fill in next array (within vertex) • Fill in vert array. The data in vert is the first subscript in tail that corresponds to the current vertex • The tail, head and next arrays have as many elements as the digraph has edges • The vert array has as many elements as the digraph has vertices

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