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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

CSCI 115

Chapter 4

Relations and Digraphs

csci 1151

CSCI 115

§4.1

Product Sets and Partitions

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 partitions1
§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
csci 1152

CSCI 115

§4.2

Relations and Digraphs

4 2 relations and digraphs
§4.2 – Relations and Digraphs
  • Relations
    • Relation from A to B
      • Subset of A x B
    • Relation on A
      • Subset of A x A
4 2 relations and digraphs1
§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
4 2 relations and digraphs2
§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.
4 2 relations and digraphs3
§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

4 2 relations and digraphs4
§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
4 2 relations and digraphs5
§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).
csci 1153

CSCI 115

§4.3

Paths in Relations and Digraphs

4 3 paths in relations and digraphs
§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
4 3 paths in relations and digraphs1
§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
4 3 paths in relations and digraphs2
§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)
4 3 paths in relations and digraphs3
§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
csci 1154

CSCI 115

§4.4

Properties of Relations

4 4 properties of relations
§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
4 4 properties of relations1
§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.
4 4 properties of relations2
§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)
4 4 properties of relations3
§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
4 4 properties of relations4
§4.4 – Properties of Relations
  • Graph of a symmetric relation
    • Undirected edge
    • Adjacent vertices
  • Connected graphs
  • Disconnected graphs
csci 1155

CSCI 115

§4.5

Equivalence Relations

4 5 equivalence relations
§4.5 – Equivalence Relations
  • Equivalence Relation
    • A relation R on a set A is an equivalence relation if it is reflexive, symmetric, and transitive
4 5 equivalence relations1
§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.
4 5 equivalence relations2
§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.
4 5 equivalence relations3
§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.
csci 1156

CSCI 115

§4.6

Data Structures for Relations and Digraphs

4 6 data structures for relations and digraphs
§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
4 6 data structures for relations and digraphs1
§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)
4 6 data structures for relations and digraphs2
§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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