Chapter 4: Matrices and Closures of Relations

1. Chapter 4:Matrices and Closures of Relations Discrete Mathematical Structures: Theory and Applications

2. 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

3. Matrices Discrete Mathematical Structures: Theory and Applications

4. Matrices Discrete Mathematical Structures: Theory and Applications

5. Matrices Discrete Mathematical Structures: Theory and Applications

6. Matrices Discrete Mathematical Structures: Theory and Applications

7. Matrices Discrete Mathematical Structures: Theory and Applications

8. 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

9. Matrices Discrete Mathematical Structures: Theory and Applications

10. Matrices Discrete Mathematical Structures: Theory and Applications

11. Matrices Discrete Mathematical Structures: Theory and Applications

12. Matrices Discrete Mathematical Structures: Theory and Applications

13. 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

14. 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

15. Discrete Mathematical Structures: Theory and Applications

16. Matrices Discrete Mathematical Structures: Theory and Applications

17. Discrete Mathematical Structures: Theory and Applications

18. 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

19. Discrete Mathematical Structures: Theory and Applications

20. 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

21. 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

22. Matrices Discrete Mathematical Structures: Theory and Applications

23. Matrices Discrete Mathematical Structures: Theory and Applications

24. Matrices Discrete Mathematical Structures: Theory and Applications

25. Discrete Mathematical Structures: Theory and Applications

26. The Matrix of a Relation and Closure Discrete Mathematical Structures: Theory and Applications

27. The Matrix of a Relation and Closure Discrete Mathematical Structures: Theory and Applications

28. The Matrix of a Relation and Closure Discrete Mathematical Structures: Theory and Applications

29. The Matrix of a Relation and Closure Discrete Mathematical Structures: Theory and Applications

30. Discrete Mathematical Structures: Theory and Applications

31. 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

32. 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

33. 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

34. Warshall’s Algorithm for Determining the Transitive Closure Discrete Mathematical Structures: Theory and Applications

35. Warshall’s Algorithm for Determining the Transitive Closure Discrete Mathematical Structures: Theory and Applications

36. Warshall’s Algorithm for Determining the Transitive Closure Discrete Mathematical Structures: Theory and Applications

37. 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