Chapter 4: Matrices and Closures of Relations

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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

Discrete Mathematical Structures:

Theory and Applications

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

Discrete Mathematical Structures: Theory and Applications

Matrices

Discrete Mathematical Structures: Theory and Applications

Matrices

Discrete Mathematical Structures: Theory and Applications

Matrices

Discrete Mathematical Structures: Theory and Applications

Matrices

Discrete Mathematical Structures: Theory and Applications

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

Matrices

Discrete Mathematical Structures: Theory and Applications

Matrices

Discrete Mathematical Structures: Theory and Applications

Matrices

Discrete Mathematical Structures: Theory and Applications

Matrices

Discrete Mathematical Structures: Theory and Applications

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

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

Matrices

Discrete Mathematical Structures: Theory and Applications

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

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

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

Matrices

Discrete Mathematical Structures: Theory and Applications

Matrices

Discrete Mathematical Structures: Theory and Applications

Matrices

Discrete Mathematical Structures: Theory and Applications

The Matrix of a Relation and Closure

Discrete Mathematical Structures: Theory and Applications

The Matrix of a Relation and Closure

Discrete Mathematical Structures: Theory and Applications

The Matrix of a Relation and Closure

Discrete Mathematical Structures: Theory and Applications

The Matrix of a Relation and Closure

Discrete Mathematical Structures: Theory and Applications

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

• 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

• 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

Discrete Mathematical Structures: Theory and Applications

Discrete Mathematical Structures: Theory and Applications

Discrete Mathematical Structures: Theory and Applications

• 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