CSCI 115

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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

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
• 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 115

§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 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 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 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 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 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 115

§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 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 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 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 115

§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 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 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 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 Relations
• Graph of a symmetric relation
• Undirected edge
• Connected graphs
• Disconnected graphs

### CSCI 115

§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 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 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 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 115

§4.6

Data Structures for Relations and Digraphs

§4.6 – Data Structures for Relations and Digraphs
• Storage cell
• Data
• Pointer
• Implementations
• Structures
• Arrays
• We will use the array implementation
§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 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