chapter iii - mapping ii

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Restriction, characteristic functions

Substitutions

Collections

Mappings - Part II

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 3
http://find/http://goback/


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Substitutions

Collections

Agenda

1 Restriction, characteristic functions

2 Substitutions

3 Collections

Collections of sets

Collections of maps

http://goforward/http://find/http://goback/


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Substitutions

Collections

Restriction and Extension

Definition

Let f : S T be a map, A be a subset of S. The restriction of f

to A is a map, denoted by f |A, from A to T given byf |A (x) = f(x) for all x A.

Definition

Let g be a restriction of f to A. Then f is called an extension of

g to X.



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Substitutions

Collections

Characteristic function

Definition

Let A X, the map A

{0,1} given by

A(x) =

1 if x A

0 if x A

is called the characteristic function of the set A.



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Substitutions

Collections

Projection

Definition

Given X = X1 X2.a) The map p1 : X1 X2 X1 defined by p1(x1, x2) = x1 is

called the canonical projection on X1.

b) The map p2 : X1 X2 X2 defined by p2(x1, x2) = x2 iscalled the canonical projection on X2.


R


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Substitutions

Collections

SubstitutionsDefinition

Definition

A bijective mapping from a finite set X into itself is called a

substitution (or permutation) of X.

Let X = {1,2, . . . , n}, f : X X be a bijective mapping. Thenwe can write

f = 1 2 . . . nf(1) f(2) . . . f(n)

where the first row contains the elements of X and the second

row contains the corresponding images.


R t i ti h t i ti f ti


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Substitutions

Collections

SubstitutionsProperties

Proposition

a) Composition of substitutions of X is also a substitution ofX .

b) The inverse map of a substitution of X is also a substitution

of X .

c) If X contains n elements then there are n! substitution of X.


Restriction characteristic functions


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Substitutions

Collections

SubstitutionsExamples

Example

Let X = {1,2,3, 4,5} and two substitutions of X as following

f = 1 2 3 4 55 3 2 1 4

,g = 1 2 3 4 53 5 1 4 2

Find substitutions g f and f1.We have

g f = 1 2 3 4 52 1 5 3 4

and

f1 =

1 2 3 4 5

4 3 2 5 1




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Substitutions

Collections

Substitutionscontinue...

Definition

A substitution is called a cycle of length k and denoted by

(i1, i2, . . . , ik) if f(i1) = i2, f(i2) = i3, . . . , f(ik) = i1 and f(j) = j forall j {i1, i2, . . . , ik}.

Example

Suppose that X = {1,2, 3,4,5} then the substitution

f =

1 2 3 4 52 4 3 1 5

is a cycle of length 3 and can be

written by f = (1, 2,4).




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Substitutions

Collections


A cycle f of length 2 is called a transposition, i.e., f = (i1, i2).

That means f(i1) = i2, f(i2) = i1 and f(j) = j for j = i1, i2.Proposition

a) Any substitution is a product of cycles.

b) Any substitution is a product of transposition.




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Substitutions

Collections


Example

a) Let X = {1, 2,3,4, 5}. We have

f = 1 2 3 4 52 4 3 1 5

= (1,2,4) = (2,4)(1, 4)Since (2,4) =

1 2 3 4 5

1 4 3 2 5

and

(1,4) =

1 2 3 4 5

4 2 3 1 5 b) Let X = {1, 2,3,4, 5,6,7}. We have

f =

1 2 3 4 5 6 7

4 3 6 5 7 2 1

= (1, 4,5,7)(2,3,6) =

(5,7)(4,7)(1,7)(3,6)(2,6)




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Substitutions

Collections


Definition

Given a substitution 1 2 . . . nf(1) f(2) . . . f(n)

a) (i,j) is called an inversion of f if i< j and f(i) > f(j)

b) f is called even if the number of inversions for f is even.

b) f is called odd if the number of inversions for f is odd.

c) If N(f) is the number of inversions of f then the sign of fdenoted by sign(f) is given by sign(f) = (1)N(f).




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,

Substitutions

Collections


Example

Let X = {1,2,3, 4,5,6, 7}. For f =

1 2 3 4 5 6 7

4 3 6 5 7 2 1

has N(f) = 12. Then f is an even substitution and sign(f) = 1.




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Substitutions

Collections


Denote Sn the set of substitutions of X = {1,2, . . . , n}.

Proposition

For f, g Sn, we have

a) sign(f g) = sign(f) sign(g)

b) sign(f) = sign(f1)

c) If f is a cycle of length k thensign(f) = (1)k+1.

d) sign(f) = 1 if f is a transposition.


Restriction, characteristic functionsCollections of sets


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Substitutions

Collections

Collections of sets

Collections of maps

Agenda


2 Substitutions

3 Collections

Collections of sets

Collections of maps




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Substitutions

Collections

Collections of sets

Collections of maps

Collections of setsDefinition

In several cases, we consider a set that its elements are sets.For example, the set of straight lines on the plane, where a line

is an element of this set but it is a set of points.

Definition

Given a set X. A set C consists of some subsets of X is calleda collections of subsets of X.

Let C is a collection of subsets of X. The intersection ofcollection C is given by

C =

AC

A = {x | x A, for all A in C}

The union of C is given by

C = AC

A = {x | x A, for some A in C}




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Substitutions

Collections

Collections of sets

Collections of maps

Collections of setsExamples

Example

Suppose that

X = {IN | n< 25}O = {n X | n is odd}

S = {n X | n is square}

P = {n X | n is prime}

C = {O,P,S}. Then

C = andC = {1,2,3,4,5,7,9, 11,13,15,16,17,19,21,23}



S b tit tiCollections of sets


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Substitutions

Collections

Collections of sets

Collections of maps

Power setDefinition

Definition

Given a set X, the power set P(X) of X is defined byP(X) = {A | A is a subset of X}.

Example

Let X= {

1}

thenP(

X) = {

,

{1

}}



S bstit tionsCollections of sets


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Substitutions

CollectionsCollections of maps

Agenda


2 Substitutions

3 Collections

Collections of sets

Collections of maps



SubstitutionsCollections of sets


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Substitutions


Collections of mapsDefinition

Definition

Let X and Y be two sets. A set whose elements are maps from

X to Y is a collection of maps from X to Y.

The collection of all maps from X to Y is denoted by

F(X,Y) = {f | f : X Y}.





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Substitutions


Collections of mapsExample

Example

If X = {x, y},Y = {1, 2} then F(X,Y) = {f1, f2, f3, f4} where

f1 : x 1, y 1,

f2 : x 1, y 2,

f3 : x 2, y 1,

f4 : x 2, y 2.





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Substitutions


Collections of mapsContinue...

PropositionLet X be a set and T = {0,1}, there is a bijective mapping fromthe power setP(X) to the collectionF(X,T).


chapter iii - mapping ii

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