Download presentation

Presentation is loading. Please wait.

Published byTrever Hays Modified over 3 years ago

1
Discrete Mathematics 3. MATRICES, RELATIONS, AND FUNCTIONS Lecture 5 Dr.-Ing. Erwin Sitompul http://zitompul.wordpress.com

2
5/2 Erwin SitompulDiscrete Mathematics 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 Homework 4

3
5/3 Erwin SitompulDiscrete Mathematics Solution: a) A (A B) = (A A) (A B) Distributive Laws = U (A B) Complement Laws = A B Identity Laws Solution of Homework 4 b) A (A B) = (A A) (A B) Distributive Laws = (A B) Complement Laws = A B Identity Laws

4
5/4 Erwin SitompulDiscrete Mathematics 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
5/5 Erwin SitompulDiscrete Mathematics The symmetric matrix is a matrix with a ij = a ji for each i and j. The zero-one (0/1) matrix is a matrix whose elements has the value of either 0 or 1. Matrices

6
5/6 Erwin SitompulDiscrete Mathematics 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. Relations

7
5/7 Erwin SitompulDiscrete Mathematics 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 R DSA Relations

8
5/8 Erwin SitompulDiscrete Mathematics 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) }. Relations

9
5/9 Erwin SitompulDiscrete Mathematics 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) }. 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. Relations

10
5/10 Erwin SitompulDiscrete Mathematics 1. Representation using arrow diagrams Representation of Relations

11
5/11 Erwin SitompulDiscrete Mathematics 2. Representation using tables Representation of Relations

12
5/12 Erwin SitompulDiscrete Mathematics 3. Representation using matrices Suppose R is a relation between A = { a 1,a 2, …,a m } and B = { b 1,b 2, …,b n }. The relation R can be presented by the matrix M = [m ij ] where: Representation of Relations

13
5/13 Erwin SitompulDiscrete Mathematics a 1 = Amir, a 2 = Budi, a 3 = Cora, and b 1 = DM, b 2 = DSA, b 3 = SP, b 4 = E3 p 1 = 2, p 2 = 3, p 3 = 4, and q 1 = 2, q 2 = 4, q 3 = 8, q 4 = 9, q 5 = 15 a 1 = 2, a 2 = 3, a 3 = 4, a 4 = 8, a 5 = 9 Representation of Relations

14
5/14 Erwin SitompulDiscrete Mathematics 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 b terminal 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. Representation of Relations

15
5/15 Erwin SitompulDiscrete Mathematics 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: Representation of Relations

16
5/16 Erwin SitompulDiscrete Mathematics Binary Relations The relations on one set is also called binary relation. A binary relation may have one or more of the following properties: 1.Reflexive 2.Transitive 3.Symmetric 4.Anti-symmetric

17
5/17 Erwin SitompulDiscrete Mathematics 1. Reflexive Relation R on set A is reflexive if (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 not reflexive because (3,3) R. Binary Relations

18
5/18 Erwin SitompulDiscrete Mathematics 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? S is not reflexive, because although (2,2) is a member of S, there exist (a,a) S for a N, such as (1,1), (3,3),.... T is not reflexive because there is even no single pair (a,a) T that can fulfill the relation. Binary Relations

19
5/19 Erwin SitompulDiscrete Mathematics If a relation is reflexive, then the main diagonal of the matrix representing it will have the value 1, or m ii = 1, for i = 1, 2, …, n. The digraph of a reflexive relation is characterized by the loop on each vertex. Binary Relations

20
5/20 Erwin SitompulDiscrete Mathematics 2. Transitive Relation R on set A is transitive if (a,b) R and (b,c) R, then (a,c) R for all a, b, c A. Binary Relations

21
5/21 Erwin SitompulDiscrete Mathematics 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. Binary Relations

22
5/22 Erwin SitompulDiscrete Mathematics 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? S is not transitive, 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) } not transitive because (3,7) R. Binary Relations

23
5/23 Erwin SitompulDiscrete Mathematics 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. 3. Symmetric and Anti-symmetric Symmetric relation Anti-symmetric relation Binary Relations

24
5/24 Erwin SitompulDiscrete Mathematics 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 not symmetric, 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. Binary Relations

25
5/25 Erwin SitompulDiscrete Mathematics 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 not symmetric, 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. Binary Relations

26
5/26 Erwin SitompulDiscrete Mathematics 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). Binary Relations

27
5/27 Erwin SitompulDiscrete Mathematics 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. Binary Relations

28
5/28 Erwin SitompulDiscrete Mathematics 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) R and 3 = 3. Binary Relations

29
5/29 Erwin SitompulDiscrete Mathematics 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? S is symmetric, because take (3,1) and (1,3) are members of S. S is not 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. Binary Relations

30
5/30 Erwin SitompulDiscrete Mathematics 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 }. Inverse of Relations

31
5/31 Erwin SitompulDiscrete Mathematics Example: Suppose P = { 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) }. Inverse of Relations

32
5/32 Erwin SitompulDiscrete Mathematics If M is a matrix representing a relation R, then the matrix representing R –1, say N, is the transpose of matrix M. N = M T, means that the rows of M becomes the columns of N Inverse of Relations

33
5/33 Erwin SitompulDiscrete Mathematics Homework 5 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.1: No.2: Represent the relation R, S, and T using matrices and digraphs.

Similar presentations

Presentation is loading. Please wait....

OK

Intracellular Compartments and Transport

Intracellular Compartments and Transport

© 2018 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on indian administrative services Ppt on airbag working principle Ppt on architecture of mughal period Ppt on rulers and buildings Ppt on vehicle chassis Ppt on ruby programming language Ppt on electronic media Ppt on applied operational research techniques Ppt on cse related topics to economics Ppt on solar system with sound