Discrete Mathematics 3. MATRICES, RELATIONS, AND FUNCTIONS Lecture 5 Dr.-Ing. Erwin Sitompul .

Discrete Mathematics

3. MATRICES, RELATIONS, AND FUNCTIONS

Lecture 5

Dr.-Ing. Erwin Sitompulhttp://zitompul.wordpress.com

5/2Erwin Sitompul Discrete Mathematics

Prove that for arbitrary sets A and B, the following set equation apply:a) A (A B) = A Bb) A (A B) = A B

Homework 4


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


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 nn. Example of a matrix, with the size of 34, is:

11 12 1

21 22 2

1 2

n

n

m m mn

a a a

a a aA

a a a

2 5 0 6

8 7 5 4

3 1 1 8

A


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.

2 6 6 4

6 3 7 3

6 7 0 2

4 3 2 8

A

0 1 1 0

0 1 1 1

0 0 0 0

1 0 0 1

A

Matrices


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


Example:Suppose A = { Amir, Budi, Cora } B = { Discrete Mathematics (DM), Data Structure and Algorithm (DSA), State Philosophy (SP), English III (E3) }

AB = { (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


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


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


1. Representation using arrow diagrams

Representation of Relations


2. Representation using tables



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:

1 11 12 1

2 21 22 2

1 2

n

n

m m m mn

a m m m

a m m mM

a m m m

1 2 nb b b

1, ( , )

0, ( , )i j

iji j

a b Rm

a b R



1 0 1 0

1 0 0 1

0 0 1 0A BM

1 1 1 0 0

0 0 0 1 1

0 1 1 0 0P QM

1 0 1 1 0

0 1 0 0 1

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

A AM

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



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.



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:



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.Reflexive2.Transitive3.Symmetric4.Anti-symmetric


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


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 = 10Are 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


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.

1

1

1

1

Binary Relations


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


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


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


S : x + y = 4, T : 3x + y = 10Are 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


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


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


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


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


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


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



S : x + y = 4, T : 3x + y = 10Are 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


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


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) }.



1 1 1 0 0

0 0 0 1 1

0 1 1 0 0P QM

If M is a matrix representing a relation R,

then the matrix representing R–1, say N, is the transpose of matrix M.

T

1 0 0

1 0 1

1 0 1

0 1 0

0 1 0

Q P P QN M

N = MT, means that the rows of M becomes the columns of N



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.

Discrete Mathematics 3. MATRICES, RELATIONS, AND FUNCTIONS Lecture 5 Dr.-Ing. Erwin Sitompul .

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