Chapter IV - Relation

8/4/2019 Chapter IV - Relation

1/32

On relation concepts

Order relation

Equivalence relation

Relations

NGUYEN CANH Nam1

1Faculty of Applied MathematicsDepartment of Applied Mathematics and Informatics

Hanoi University of [email protected]

HUT - 2010

NGUYEN CANH Nam Mathematics I - Chapter 4
http://-/?-http://-/?-http://goforward/http://find/http://goback/http://-/?-


2/32


Order relation


Agenda

1 On relation concepts

2 Order relation

Concepts on order relationLexicographical order

3 Equivalence relation

Definitions and examples

Equivalence classes

Partitions induced by maps

http://-/?-http://-/?-http://find/http://goback/http://-/?-


3/32


Order relation


RelationDefinition

Definition

Let A and B be sets, and let R be a subset of A B. Then R iscalled a relation from A to B.

In other words, a binary relation from A to B is a set R ofordered pairs where the first element of each ordered pair

comes from A and the second elements comes from B.

If (x, y) R, then x is said to be in relation R to y, written xRy.A relation from A to A is called a relation on A (or in A).

http://find/http://goback/http://-/?-


4/32


Order relation


RelationExamples

Example

Denote by X the set of all inhabitances of some island.

Let U be the subset of X X given by (a, b) U iff a is theuncle of b. Then U is a relation on X.

Let N be the subset of X X given by (x, y) N iff x is theniece of y. Then N is also a relation of X.

ExampleX is the set of of real numbers. The subset S X X given by(a, b) S iff a b. Then S or is a relation on X.



5/32


Order relation


RelationExamples (continue...)

Example

Let A be the set of students in your school, and let B be the set

of courses. A relation R can be defined by pairs (a, b), where ais a student enrolled in course b.

Example

Let A be the set of all districts, and let B be the set of the all

provincials in Viet Nam. Define the relation R by (a, b) belongsto R if district a is in provincial b.



6/32


Order relation


RelationProperties

Definition

Let R be a relation in the set X. R is said to be reflexive if xRxfor all x X

Example

Is the "divides" relation on the set of positive integers reflexive?

Because a | awhenever a is a positive integer, the "divides"

relation is reflexive. (Note that if we replace the set of positiveintegers with the set of all integers the relation is not reflexive

because 0 does not divide 0.)


O l ti t


7/32


Order relation


RelationProperties (continue...)

Definition

Let R be a relation in the set X. R is said to be

a) Symmetric if xRy then yRx for all x, y X

b) Antisymmetric if xRy and yRx imply x = y for allx, y X.

Example

Is the "divides" relation on the set of positive integers

symmetric? Is it antisymmetric?

This relation is not symmetric because 1 | 2, but 2 \| 1. It isantisymmetric, for if aand b are positive integers with a | b andb | a, then a= b (the verification of this is left as an exercise).




8/32


Order relation


RelationProperties (continue...)

Definition

Let R be a relation in the set X. R is said to be transitive if

xRy and yRz imply xRz for all x, y, z X

Example

Is the "divides" relation on the set of positive integers transitive?

Suppose that adivides b and b divides c. Then there are

positive integers k and I such that b = ak and c = bl. Hence,c = a(kl), so adivides c. It follows that this relation is transitive.




9/32


Order relation


Concepts on order relation

Lexicographical order

Agenda


2 Order relation




Equivalence classes



http://-/?-http://-/?-http://find/http://goback/http://-/?-


10/32


Order relation




Order relationDefinitions

Definition

A relation R on X is called an (partial) oder relation if R is

reflexive, antisymmetric and transitive. An order relation isusually denoted by . That means is an order relation if

i) a a for all a X.

ii) If a b and b a then a= b.

iii) If a

b and b

c then a

c.

An order relation on X is called total order if for all a, b in Xeither a b or b a.


http://goforward/http://find/http://goback/http://-/?-


11/32


Order relation




Order relationExamples

Example

Let X be a set whose elements are themselves sets. Consider

the relation determined by "set inclusion". For any sets

A, B, C X we see that(i) A A;

(ii) if A B and B A, then A = B;

(iii) if A B and B C, then A C.

Hence, set inclusion is reflexive, antisymmetric and transitive;therefore it is a partial order in X.

For any two set A and B, we may not have A B neitherB A. So set inclusion is not a total order relation.


On relation conceptsC d l i


12/32


Order relation




Order relationExamples (continue...)

Example

Consider the relation ("less than or equal to") on the set IR ofreal numbers, we have

(i) a a;

(ii) if a b and b a, then a= b;

(iii) if a b and b c, then a c.

Hence this relation is reflexive, transitive and antisymmetric;

therefore, it is a partial order on IR.

Moreover, we can always compare two real numbers. So the

relation is a total order relation on IR.


On relation conceptsC t d l ti


13/32

p

Order relation




Order relationUpper bounds, lower bounds, greatest element, smallest element

Definition

Given an order relation on X. Let A X .

a) If x X such that a x for all a A then x is called anupper bound of A.

b) If y X such that y a for all a A then y is called alower bound of A.

c) An element x0 is called the greatest element of A if x0 A

and x0 is an upper bound of A.

d) An element y0 is called the smallest element of A if y0 Aand y0 is a lower bound of A.


On relation conceptsConcepts on order relation


14/32

p

Order relation





Example

Consider the relation ("less than or equal to") on the set IR of

real numbers and A = [1; 5], B = (1; 5).6 is an upper bound of A, 7 is an upper bound of B.

0 is a lower bound of A, 0.9 is a lower bound of B.

5 is the greatest element of A, there does not exists the

greatest element of B.

1 is the smallest element of A, there does not exists the

smallest element of B.




15/32

Order relation





Remark

1 The smallest element of A is unique,

2 The greatest element of A is unique.

3 In general there does not exist always the smallest and the

greatest element.




16/32

Order relation




Order relationMaximal, minimal

Definition

Let be an order relation on X and S X. An element x Sis called a maximal element of S if for a S , x a impliesx = a. An element y S is called a minimal element of S iffor a S, a y implies y = a.




17/32

Order relation





Example

Consider the set X IR2 defined as in the picture below.



O d l tiConcepts on order relation


18/32

Order relation





Example (continue)

We define a relation on X as follows

(a, b) (c, d) (a c) (b d)

It is an order relation (Verify!)

So

(1

2

, 1

2

) is a minimal element of X, (1

3

, 2

3

) is also a

minimal element of X.

(1

2,

1

2) is a maximal element of X, (

1

4,

3

4) is also a maximal

element of X.



Order relationConcepts on order relation


19/32

Order relation


p


Agenda


2 Order relation




Equivalence classes






20/32

Order relation


p


Lexicographical orderDefinition

Definition

Given a total order relation on X. We define an order relationon Xn as follows

(x1, . . . , xn) < (y1, . . . , yn) if there is an index k, 1 k n 1

such that xi = yi i k and xk+1 yk+1, and

(x1, . . . , xn) (y1, . . . , yn) if (x1, . . . , xn) = (y1, . . . , yn)

or (x1, . . . , xn) < (y1, . . . , yn).

This relation on Xn is a total order relation and is called thelexicographical relation on Xn.





21/32

Order relation

Equivalence relationLexicographical order

Lexicographical orderExample

Example

Given X = {0, 1} with the total order as 0 0, 0 1, 1 1.In the lexicographical order, compare elements

x = (1, 1, 0, 1, 0, 1, 1), y = (1, 0, 1, 1, 0, 0, 0) andz = (1, 0, 1, 0, 1, 1, 1).Solution.

y < x because y1 = x1 and y2 < x2.

z < y because z1 = y1, z2 = y2 and z3 < y3.

z < x by using the transitive property of the order relation.



Order relation


Equivalence classes


22/32

Order relation


Equivalence classes


Agenda


2 Order relation




Equivalence classes



On relation conceptsOrder relation

Definitions and examplesEquivalence classes
http://-/?-http://-/?-http://-/?-http://-/?-http://goforward/http://find/http://goback/http://-/?-


23/32

Order relation


Equivalence classes


Equivalence relationDefinition

Definition

A relation R on X is called an equivalence relation if R is

reflexive, symmetric and transitive. An equivalence relation isusually denoted by :. That means : is an equivalence relation if

i) a : a for all a X.

ii) If a : b then b : a.

iii) If a : b and b : c then a : c.





24/32


q


Equivalence relationExamples

Example

Let X be the set of all students of HUT. Define xRy if x and y

are in the same class. Then R is an equivalence relation.

Example

Let X be the set of all lines in a plane. For x, y X let xymean that x is parallel to y. Let us further agree that every line

is parallel to itself. Then is an equivalence relation on X.Similarly, similarity of triangle is an equivalence relation.





25/32


q


Equivalence relationExamples(continue...)

Example

The relation congruence modulo n on ZZ is defined as follows.

Let n be a fixed positive integer. For any x, y ZZ, x is said tobe congruent to y (modulo n), written

x y(mod n)

if n divides x y. Now for any x, y, z in ZZ, it is true that

(i) n divides x x = 0; hence x x(mod n);

(ii) if n divides x y, then n divides y x; (!!!!!!!)

(iii) if n divides x y and also y z, then n divides x z.

This proves that congruence modulo n is an equivalence

relation on ZZ.





26/32

Equivalence relation Partitions induced by maps

Agenda


2 Order relation




Equivalence classesPartitions induced by maps



http://-/?-http://-/?-http://-/?-http://-/?-http://goforward/http://find/http://goback/http://-/?-http://-/?-http://-/?-


27/32


Equivalence classesDefinition

Definition

Let E be an equivalence relation on a set X, and let a X. Theset of all elements in X that are in relation E to a is called theequivalence class of aunder E and is denoted by E(a). That is,

E(a) = {x X | xEa}

A subset C of X is called an equivalence class of E in X ifC = E(a) for some a in X. The set of all equivalence classes of

E in X is called the quotient set of X by E and is written X/E.That is,

X/E = {C X | C = E(a) for some a X}



http://-/?-http://-/?-http://goforward/http://find/http://goback/http://-/?-http://-/?-http://-/?-http://-/?-


28/32


Equivalence classesExample

Example

Let E be a relation on the set of integer numbers ZZ such that

xEy if x y is a multiple of 4. We can easily prove that E is anequivalence relation. The equivalence classes are

E(0) = {z = 4k | k ZZ}

E(1) = {z = 4k + 1 | k ZZ}

E(2) = {z = 4k + 2 | k ZZ}

E(3) = {z = 4k + 3 | k ZZ}



E i l l i


P i i i d d b
http://-/?-http://-/?-http://goforward/http://find/http://goback/http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-


29/32


Equivalence classesResults

Proposition

LetE be an equivalence relation on X. We have

i) If y E(x) thenE(y) = E(x),

ii) Two equivalence classes are either distinct or equal.

Proposition

LetE be an equivalence relation on X. Then the collection of all equivalence

classes is a partition on X.

RemarkA partition of a set X induces an equivalence relation on X. Actually, if

X =

iI

Ai is a partition (Ai Ai = ). We define a relation: on X as follows.

For x, y X, x : y if x, y are in the same subset Ai. We can easily check that: is an equivalence relation.



E i l l ti


P titi i d d b


30/32


Agenda


2 Order relation




Equivalence classesPartitions induced by maps





http://-/?-http://-/?-http://-/?-http://-/?-http://goforward/http://find/http://goback/http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-


31/32



Let f : X Y be a map. then f induces an equivalence relationon X as follows.

Proposition

Let f be a map from X to Y . The relationR on X is defined byaRb if f(a) = f(b). ThenR is an equivalence relation on X.

Remark

From the above proposition a map from X to Y induces anequivalence relation on X and therefore a partition on X.







32/32


Partitions induced by mapsExamples

Example

a) Let f : IR IR be a map defined by f(x) = x2. Thecorresponding equivalence relation : defined by

x : y if |x| = |y|.

b) Consider the map f : ZZ ZZ defined by f(n) = (1)n. Thecorresponding equivalence relation : then defined by

x : y if x y is even.

The corresponding partition is ZZ = Z1 Z2 whereZ1 = {2n+ 1 | n ZZ} and Z2 = {2n | n ZZ}

http://-/?-http://-/?-http://goforward/http://find/http://goback/http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-

Date post:	07-Apr-2018
Category:	Documents
Upload:	mai-thanh-tung
View:	226 times
Download:	0 times

Chapter IV - Relation

Documents