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Chapter 4: Matrices and Closures of Relations. Discrete Mathematical Structures: Theory and Applications. Learning Objectives. Learn about matrices and their relationship with relations Become familiar with Boolean matrices

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chapter 4 matrices and closures of relations

Chapter 4:Matrices and Closures of Relations

Discrete Mathematical Structures:

Theory and Applications

learning objectives
Learning Objectives
  • Learn about matrices and their relationship with relations
  • Become familiar with Boolean matrices
  • Learn the relationship between Boolean matrices and different closures of a relation
  • Explore how to find the transitive closure using Warshall’s algorithm

Discrete Mathematical Structures: Theory and Applications

matrices
Matrices

Discrete Mathematical Structures: Theory and Applications

matrices4
Matrices

Discrete Mathematical Structures: Theory and Applications

matrices5
Matrices

Discrete Mathematical Structures: Theory and Applications

matrices6
Matrices

Discrete Mathematical Structures: Theory and Applications

matrices7
Matrices

Discrete Mathematical Structures: Theory and Applications

matrices8
Matrices
  • Two matrices are added only if they have the same number of rows and the same number of columns
  • To determine the sum of two matrices, their corresponding elements are added

Discrete Mathematical Structures: Theory and Applications

matrices9
Matrices

Discrete Mathematical Structures: Theory and Applications

matrices10
Matrices

Discrete Mathematical Structures: Theory and Applications

matrices11
Matrices

Discrete Mathematical Structures: Theory and Applications

matrices12
Matrices

Discrete Mathematical Structures: Theory and Applications

matrices13
Matrices
  • The multiplication AB of matrices A and B is defined only if the number of rows and columns of A is the same as the number of rows and of B

Discrete Mathematical Structures: Theory and Applications

matrices14
Matrices

Figure 4.1

  • Let A = [aij]m×nbe an m × n matrix and B = [bjk]n×pbe an n × p matrix. Then AB is defined
  • To determine the (i, k)th element of AB, take the ith row of A and the kth column of B, multiply the corresponding elements, and add the result
  • Multiply corresponding elements as in Figure 4.1

Discrete Mathematical Structures: Theory and Applications

matrices16
Matrices

Discrete Mathematical Structures: Theory and Applications

matrices18
Matrices
  • The rows of A are the columns of ATand the columns of A are the rows of AT

Discrete Mathematical Structures: Theory and Applications

matrices20
Matrices
  • Boolean (Zero-One) Matrices
    • Matrices whose entries are 0 or 1
    • Allows for representation of matrices in a convenient way in computer memory and for design and implement algorithms to determine the transitive closure of a relation

Discrete Mathematical Structures: Theory and Applications

matrices21
Matrices
  • Boolean (Zero-One) Matrices
    • The set {0, 1} is a lattice under the usual “less than or equal to” relation, where for all a, b ∈ {0, 1}, a ∨ b = max{a, b} and a ∧ b = min{a, b}

Discrete Mathematical Structures: Theory and Applications

matrices22
Matrices

Discrete Mathematical Structures: Theory and Applications

matrices23
Matrices

Discrete Mathematical Structures: Theory and Applications

matrices24
Matrices

Discrete Mathematical Structures: Theory and Applications

the matrix of a relation and closure
The Matrix of a Relation and Closure

Discrete Mathematical Structures: Theory and Applications

the matrix of a relation and closure27
The Matrix of a Relation and Closure

Discrete Mathematical Structures: Theory and Applications

the matrix of a relation and closure28
The Matrix of a Relation and Closure

Discrete Mathematical Structures: Theory and Applications

the matrix of a relation and closure29
The Matrix of a Relation and Closure

Discrete Mathematical Structures: Theory and Applications

slide31
ALGORITHM 4.3: Compute the transitive closure
  • Input: M —Boolean matrices of the relation R n—positive integers such that n × n specifies the size of M
  • Output: T —an n × n Boolean matrix such that T is the transitive closure of M
    • 1. procedure transitiveClosure(M,T,n)
    • 2. begin
    • 3. A := M;
    • 4. T := M;
    • 5. for i := 2 to n do
    • 6. begin
    • 7. A := //A = Mi
    • 8. T := T ∨ A; //T= M ∨ M2∨ · · · ∨ Mi
    • 9. end
    • 10. end

Discrete Mathematical Structures: Theory and Applications

warshall s algorithm for determining the transitive closure
Warshall’s Algorithm for Determining the Transitive Closure
  • Previously,the transitive closure of a relation R was foundby computing the matrices and then taking the Boolean join
    • This method is expensive in terms of computer time
  • Warshall’s algorithm: an efficient algorithm to determine the transitive closure

Discrete Mathematical Structures: Theory and Applications

warshall s algorithm for determining the transitive closure33
Warshall’s Algorithm for Determining the Transitive Closure
  • Let A = {a1, a2, . . . , an} be a finite set, n ≥ 1, and let R be a relation on A.
  • Warshall’s algorithm determines the transitive closure by constructing a sequence of n Boolean matrices

Discrete Mathematical Structures: Theory and Applications

warshall s algorithm for determining the transitive closure34
Warshall’s Algorithm for Determining the Transitive Closure

Discrete Mathematical Structures: Theory and Applications

warshall s algorithm for determining the transitive closure35
Warshall’s Algorithm for Determining the Transitive Closure

Discrete Mathematical Structures: Theory and Applications

warshall s algorithm for determining the transitive closure36
Warshall’s Algorithm for Determining the Transitive Closure

Discrete Mathematical Structures: Theory and Applications

warshall s algorithm for determining the transitive closure37
Warshall’s Algorithm for Determining the Transitive Closure
  • ALGORITHM 4.4: Warshall’s Algorithm
  • Input: M —Boolean matrices of the relation R
  • n—positive integers such that n × n specifies the size of M
  • Output: W —an n × n Boolean matrix such thatW is the transitive closure of M
    • 1. procedure WarshallAlgorithm(M,W,n)
    • 2. begin
    • 3. W := M;
    • 4. for k := 1 to n do
    • 5. for i := 1 to n do
    • 6. for j := 1 to n do
    • 7. if W[i,j] = 1 then
    • 8. if W[i,k] = 1 and W[k,j] = 1 then
    • 9. W[i,j] := 1;
    • 10. end

Discrete Mathematical Structures: Theory and Applications