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Chapter 4: Matrices and Closures of Relations

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Chapter 4:Matrices and Closures of Relations

Discrete Mathematical Structures:

Theory and Applications

- 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

Discrete Mathematical Structures: Theory and Applications

Discrete Mathematical Structures: Theory and Applications

Discrete Mathematical Structures: Theory and Applications

Discrete Mathematical Structures: Theory and Applications

Discrete Mathematical Structures: Theory and Applications

- 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

Discrete Mathematical Structures: Theory and Applications

Discrete Mathematical Structures: Theory and Applications

Discrete Mathematical Structures: Theory and Applications

Discrete Mathematical Structures: Theory and Applications

- 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

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

Discrete Mathematical Structures: Theory and Applications

Discrete Mathematical Structures: Theory and Applications

Discrete Mathematical Structures: Theory and Applications

- The rows of A are the columns of ATand the columns of A are the rows of AT

Discrete Mathematical Structures: Theory and Applications

Discrete Mathematical Structures: Theory and Applications

- 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

- 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

Discrete Mathematical Structures: Theory and Applications

Discrete Mathematical Structures: Theory and Applications

Discrete Mathematical Structures: Theory and Applications

Discrete Mathematical Structures: Theory and Applications

Discrete Mathematical Structures: Theory and Applications

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 4.3: 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