1 / 33

Discrete Mathematics

Discrete Mathematics. 3. MATRICES, RELATIONS, AND FUNCTIONS. Lecture 5. Dr.-Ing. Erwin Sitompul. http://zitompul.wordpress.com. Homework 4. Prove that for arbitrary sets A and B , the following set equation apply: a) A  ( A  B ) = A  B b) A  ( A  B ) = A  B.

terri
Télécharger la présentation

Discrete Mathematics

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Discrete Mathematics 3. MATRICES, RELATIONS, AND FUNCTIONS Lecture5 Dr.-Ing. Erwin Sitompul http://zitompul.wordpress.com

  2. Homework 4 Prove that for arbitrary sets A and B, the following set equation apply: a) A(A  B) = AB b) A (A  B) = A B

  3. Solution of Homework 4 Solution: a) A(A  B) = (AA)  (AB) Distributive Laws = U (AB)Complement Laws = AB Identity Laws b) A(A  B) = (AA) (AB) Distributive Laws = (AB)Complement Laws = AB Identity Laws

  4. Matrices • A matrix is a structure of scalar elements in rows and columns. • The size of a matrix A is described by the number of rows m and the number of columns n, (m,n). • The square matrix is a matrix with the size of nn. • Example of a matrix, with the size of 34, is:

  5. Matrices • The symmetric matrix is a matrix with aij = aji for each i and j. • The zero-one (0/1) matrix is a matrix whose elements has the value of either 0 or 1.

  6. Relations • A binary relation R between set A and set B is an improper subset of A  B. • Notation: R (AB) • a R b is the notation for (a,b)  R, with the meaning “relation R relates a with b.” • a R b is the notation for (a,b) R, with the meaning “relation R does not relate a with b.” • Set A is denoted as the domain of R.Set B is denoted as the range of R.

  7. Relations • Example: • Suppose A = { Amir, Budi, Cora } B = { Discrete Mathematics (DM), Data Structure and Algorithm (DSA), State Philosophy (SP), English III (E3) } • AB = { (Amir,DM), (Amir, DSA), (Amir,SP), (Amir,E3), (Budi,DM), (Budi, DSA), (Budi,SP), (Budi,E3), (Cora,DM), (Cora, DSA), (Cora,SP), (Cora,E3) } • Suppose R is a relation that describes the subjects taken by a certain IT students in the May-August semester, that is: • R = { (Amir,DM), (Amir, SP), (Budi,DM), (Budi,E3), (Cora,SP) } • It can be seen that: • R (AB) • A is the domain of R, B is the range of R • (Amir,DM)  R or Amir R DM • (Amir,DSA) R or Amir RDSA

  8. Relations Example: Take P = { 2,3,4 } Q = { 2,4,8,9,15 } If the relation R from P to Q is defined as: (p,q)  R if p is the factor of q, then the followings can be obtained: R = { (2,2),(2,4),(2,8),(3,9),(3,15),(4,4),(4,8) }.

  9. Relations • The relation on a set is a special kind of relation. • That kind of relation on a set A is a relation of AA. • The relation on the set A is a subset of AA. Example: Suppose R is a relation on A = { 2,3,4,8,9 } which is defined by (x,y)  R if x is the prime factor of y, then we can obtain the relation: R = { (2,2),(2,4),(2,8),(3,3),(3,9) }.

  10. Representation of Relations 1. Representation using arrow diagrams

  11. Representation of Relations 2. Representation using tables

  12. Representation of Relations 3. Representation using matrices • Suppose R is a relation between A = {a1,a2, …,am} and B = {b1,b2, …,bn}. • The relation R can be presented by the matrix M = [mij] where:

  13. Representation of Relations a1 = Amir, a2 = Budi, a3 = Cora, and b1 = DM, b2 = DSA, b3 = SP, b4 = E3 p1 = 2, p2 = 3, p3 = 4, and q1 = 2, q2 = 4, q3 = 8, q4 = 9, q5 = 15 a1 = 2, a2 = 3, a3 = 4, a4 = 8, a5 = 9

  14. Representation of Relations 4. Representation using directed graph (digraph) • Relation on one single set can be represented graphically by using a directed graph or digraph. • Digraphs are not defined to represent a relation from one set to another set. • Each member of the set is marked as a vertex (node), and each relation is denoted as an arc (bow). • If (a,b) R, then an arc should be drawn from vertex a to vertex b. Vertex a is called initial vertex while vertex bterminal vertex. • The pair of relation (a,a) is denoted with an arch from vertex a to vertex a itself. This kind of arc is called a loop.

  15. Representation of Relations Example: Suppose R = { (a,a),(a,b),(b,a),(b,c),(b,d),(c,a),(c,d),(d,b) } is a relation on a set { a,b,c,d }, then R can be represented by the following digraph:

  16. Binary Relations • The relations on one set is also called binary relation. • A binary relation may have one or more of the following properties: • Reflexive • Transitive • Symmetric • Anti-symmetric

  17. Binary Relations 1. Reflexive • Relation Ron set Ais reflexiveif (a,a) R for each aA. • Relation R on set A is not reflexive if there exists aA such that (a,a) R. Example: Suppose set A = {1,2,3,4}, and a relation R is defined on A, then: (a) R = {(1,1),(1,3),(2,1),(2,2),(3,3),(4,2),(4,3),(4,4) } is reflexive because there exist members of the relation with the form (a,a) for each possible a, namely(1,1),(2,2),(3,3), and (4,4). (b) R = {(1,1),(2,2),(2,3),(4,2),(4,3),(4,4) } is notreflexive because (3,3) R.

  18. Binary Relations Example: Given a relation “divide without remainder” for a set of positive integers, is the relation reflexive or not? Each positive integer can divide itself without remainder (a,a)R for each a A  the relation is reflexive Example: Given two relations on a set of positive integers N: S : x + y = 4, T : 3x + y = 10 Are S and T reflexive or not? Sis notreflexive, because although (2,2) is a member of S, there exist (a,a) S for aN, such as (1,1), (3,3), .... Tis notreflexive because there is even no single pair (a,a) Tthat can fulfill the relation.

  19. Binary Relations • If a relation is reflexive, then the main diagonal of the matrix representing it will have the value 1, or mii = 1, for i = 1, 2, …, n. • The digraph of a reflexive relation is characterized by the loop on each vertex.

  20. Binary Relations 2. Transitive • Relation R on set Ais transitiveif (a,b) R and (b,c) R, then (a,c) R for all a, b, cA.

  21. Binary Relations Example: Suppose A = { 1, 2, 3, 4 }, and a relation R is defined on set A, then: (a) R = { (2,1),(3,1),(3,2),(4,1),(4,2),(4,3) } is transitive. (b) R = { (1,1),(2,3),(2,4),(4,2) } is not transitive because (2,4) and (4,2) R, but (2,2) R, also (4,2) and (2,3) R, but (4,3) R. (c) R = { (1,2), (3,4) } is transitive because there is no violation against the rule { (a,b) R and (b,c) R }  (a,c) R. Relation with only one member such as R = { (4,5) } is always transitive.

  22. Binary Relations Example: Is the relation “divide without remainder” on a set of positive integers transitive or not? It is transitive. Suppose that a divides b without remainder and b divides c without remainder, then certainly a divides c without remainder. { a R b  b R c }  a R c Example: Given two relations on a set of positive integers N: S : x + y = 4, T : 3x + y = 10 Are S and T transitive or not? Sis nottransitive, because i.e., (3,1) and (1,3) are members of S, but (3,3) and (1,1) are not members of S. T= { (1,7),(2,4),(3,1) }  nottransitive because (3,7) R.

  23. Binary Relations 3. Symmetric and Anti-symmetric • Relation R on set A is symmetric if (a,b)  R, then (b,a)  R for all a,b A. • Relation R on set A is not symmetric if there exists (a,b)  R such that (b,a)  R. • Relation R on set A such that if (a,b)  R and (b,a)  R then a = b for a,b A, is called anti-symmetric. • Relation R on set A is not anti-symmetric if there exist different a and b such that (a,b)  R and (b,a)  R. Symmetric relation Anti-symmetric relation

  24. Binary Relations Example: Suppose A = { 1,2,3,4 }, and relation R is defined on set A, then: (a) R = { (1,1),(1,2),(2,1),(2,2),(2,4),(4,2),(4,4) } is symmetric, because if (a,b)  R then (b,a)  R also . Here, (1,2) and (2,1)  R, as well as (2,4) and (4,2)  R. is not anti-symmetric, because i.e., (1,2)  R and (2,1)  R while 1  2. (b) R = { (1,1),(2,3),(2,4),(4,2) } is notsymmetric, because (2,3) R, but (3,2) R. is not anti-symmetric, because there exists (2,4)  R and (4,2)  R while 2  4.

  25. Binary Relations Example: Suppose A = { 1,2,3,4 }, and relation R is defined on set A, then: (c) R = { (1,1),(2,2),(3,3) } is symmetric and anti-symmetric, because (1,1) R and 1 = 1, (2,2) R and 2 = 2, and (3,3) R and 3 = 3. (d) R = { (1,1),(1,2),(2,2),(2,3) } is notsymmetric, because (2,3) R, but (3,2) R. is anti-symmetric, because (1,1) R and 1 = 1 and, (2,2) R and 2 = 2.

  26. Binary Relations Example: Suppose A = { 1,2,3,4 }, and relation R is defined on set A, then: (e) R = { (1,1),(2,4),(3,3),(4,2) } is symmetric. is not anti-symmetric, because there exist (2,4) and (4,2) as member of R while2  4. (f)R = { (1,2),(2,3),(1,3) } is not symmetric. is anti-symmetric, because there is no different a and b such that (a,b)  R and (b,a)  R (which will violate the anti-symmetric rule).

  27. Binary Relations Relation R = { (1,1),(2,2),(2,3),(3,2),(4,2),(4,4)} is not symmetric and not anti-symmetric. R is not symmetric, because (4,2) R but (2,4) R. R is not anti-symmetric,because (2,3) R and (3,2) R but 2  3.

  28. Binary Relations Example: Is the relation “divide without remainder” on a set of positive integers symmetric? Is it anti-symmetric? It is not symmetric, because if a divides b without remainder, then b cannot divide a without remainder, unless if a = b. For example, 2 divides 4 without remainder, but 4 cannot divide 2 without remainder. Therefore, (2,4) R but (4,2) R. It is anti-symmetric, because if a divides b without remainder, and b divides a without remainder, then the case is only true for a = b. For example, 3 divides 3 without remainder, then (3,3) Rand 3 = 3.

  29. Binary Relations Example: Given two relations on a set of positive integers N: S : x + y = 4, T : 3x + y = 10 Are S and T symmetric? Are they anti-symmetric? Sissymmetric, because take (3,1) and (1,3) are members of S. Sisnot anti-symmetric, because although there exists(2,2) R, but there exist also { (3,1),(1,3) } R while 3  1. T= { (1,7),(2,4),(3,1) }  not symmetric. T= { (1,7),(2,4),(3,1) }  anti-symmetric.

  30. Inverse of Relations If R is a relation from set A to set B, then the inverse of relation R, denoted with R–1, is the relation from set B to set A defined by: R–1 = { (b,a) | (a,b) R }.

  31. Inverse of Relations Example: SupposeP = { 2,3,4 } Q = { 2,4,8,9,15 }. If the relation R from P to Q is defined by: (p,q)  R if p divides q without remainder, then the members of the relation can be obtained as: R = { (2,2),(2,4),(2,8),(3,9),(3,15),(4,4),(4,8) }. R–1, the inverse of R, is a relation from Q to P with: (q,p) R–1 if q is a multiplication of p. It can be obtained that: R–1 = { (2,2),(4,2),(8,2),(9,3),(15,3),(4,4),(8,4) }.

  32. Inverse of Relations If M is a matrix representing a relation R, then the matrix representing R–1, say N, is the transpose of matrix M. N = MT, means that the rows of M becomes the columns of N

  33. Homework 5 No.1: For each of the following relations on set A = { 1,2,3,4 }, check each of them whether they are reflexive, transitive, symmetric, and/or anti-symmetric: (a) R = { (2,2),(2,3),(2,4),(3,2),(3,3),(3,4) } (b) S = { (1,1),(1,2),(2,1),(2,2),(3,3),(4,4) } (c) T = { (1,2),(2,3),(3,4) } No.2: Represent the relation R, S, and T using matrices and digraphs.

More Related