Chapter III - Mapping I

8/4/2019 Chapter III - Mapping I

1/42

Basic concepts

Injective, surjective, bijective mappings

Composition of maps, inverse maps

Mappings

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


2/42

Basic concepts



Agenda

1 Basic concepts

2 Injective, surjective, bijective mappings

Injective mappingSurjective mapping

Bijective mapping

3 Composition of maps, inverse maps

Composition of maps

Inverse maps



3/42

Basic concepts



MappingsDefinition

Definition

Let S, T be sets; a mapping (or a map or a function) from S

to T is a rule that assigns to each element s S a uniqueelement t T.This means that if s is a given element of S, then there is only

one element t T that is associated to s by the mapping.Denote f is a mapping from S to T by f : S

T and write

t = f(s).S is called the domain and T is called the codomain.

The symbols f(s) are read f of s".

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


4/42

Basic concepts



Mappingscontinue...

We have constantly encountered mapping often in the form of

formulas. But mappings need not be restricted to sets of

numbers. They can occur in any area.

Example1 Let S = {all men who have ever lived} and

T = {all women who have ever lived}.Define f : S T by f(s) = mother of s. Therefore

f(John F. Kennedy) = Rose Kenendy,

and according to our definition, Rose Kennedy is the image

under f of John F. Kennedy.



5/42

Basic concepts



Mappingscontinue...

Example

2 Let S be the set of all objects for sale in Vincom and let

T = {all real numbers}. Define f : S T by f(s) = priceof s. This defines a mapping from S to T.

3 Let S = {all legally employed citizens of Viet Nam} andT = {positive integers}. Define, for s S, f(s) byf(s) = ID card number of s. Then f defines a mappingfrom S to T.

4 Let S be the set of the colors of the rainbow. Let

T = {1, 2, 3, 4, 5, 6, 7}. Define f : S T by f(s) = thenumber of letters in a given color of the rainbow. Then f

defines a mapping from S to T.


B


6/42

Basic concepts



Mappingscontinue...

Definition

The mapping iX : X X given by iX(x) = x for every x X iscalled the identity on X.

DefinitionLet f be a mapping from S to T. For s S, f(s) is called theimage of s under f, also the value of f at s. The set of all

elements f(s), when s ranges over all elements of S, is calledthe image of S under f or the range of f, and is denoted by

f(S):f(S) := {t | t = f(s), s S}

or

f(S) :=

{t

| s

S, t = f(s)

}NGUYEN CANH Nam Mathematics I - Chapter 3

B i t


7/42

Basic concepts



Mappingscontinue...

Example

Denote IN the set of strictly positive integers,

IN = IN \ {0}

1 The mapping f : IN {1, 1} given by

f(n) = (1)n (n IN)

has domain IN, codomain and range {1, 1}


Basic concepts


8/42

Basic concepts



Mappingscontinue...

Example

2 The mapping f : IR IR given by

f(x) = x

2

+ 1 (x IR)has domain IR, codomain IR and range {y | y 1}.

3 The mapping f : IN OQ given by

f(x) = 1n (n IN)

has domain IN, codomain OQ and range {12

, 1,3

2, 2, ...}.


Basic concepts


9/42

Basic concepts



Mappingscontinue...

Definition

Given a mapping f : S T and a subset A T, we may wantto look at B = {s S | f(s) A}, we use the notation f1(A) forthis set B, and call f1(A) the inverse image of A under f.

Consider the subset {t} in T, if the inverse image of {t}consists of only one element, say s S, we could try to definef1(t) by defining f1(t) = s.


Basic concepts Injective mapping


10/42

Basic concepts



Injective mapping

Surjective mapping

Bijective mapping

Agenda

1 Basic concepts



Bijective mapping


Composition of maps

Inverse maps




11/42

Basic concepts



Injective mapping

Surjective mapping

Bijective mapping

Injective mappingDefinition

Definition

A mapping f : S T is said to be one-to-one (written 1-1) orinjective if for s1

= s2 in S, f(s1)

= f(s2) in T. Equivalently, f is

1-1 iff(s1) = f(s2) implies that s1 = s2.

A mapping is 1-1 if it takes distinct objects into distinct images.

In the examples we gave earlier,

The first mapping of is not 1-1(two brothers could have the

same mother).

The second mapping is 1-1 (distinct vietnamese citizens

have distinct ID card number).




12/42

Basic concepts



Injective mapping

Surjective mapping

Bijective mapping

Injective mappingExamples

Example

Figure: A injective function




13/42

p



j pp g

Surjective mapping

Bijective mapping

Injective mappingExamples

Example

The function f : IR IR defined by f(x) = 2x + 1 isinjective,

Given arbitrary real numbers x and x, if 2x + 1 = 2x + 1,then 2x = 2x, so x = x.

The function g : IR IR defined by g(x) = x2 is notinjective,

g(1) = 1 = g(1).The function h : [0, ) IR with the same formula as g isinjective.

Given arbitrary nonnegative real numbers x and x, if

x2 = x2, then |x| = |x|, so x = x.




14/42

p



j pp g

Surjective mapping

Bijective mapping

Agenda

1 Basic concepts



Bijective mapping


Composition of maps

Inverse maps




15/42



Surjective mapping

Bijective mapping

Surjective mappingDefinition

Definition

The mapping f : S

T is onto or surjective if every t

T is

the image under f of some s S; that is, if and only if, givent T, there exists an s S such that t = f(s).The mapping f : S T is onto by saying that f(S) = T.In the examples we gave earlier, the first mapping is not onto,

since not every woman that ever lived was the mother of a malechild.




16/42



Surjective mapping

Bijective mapping

Surjective mappingExamples

Example

Figure: A surjective function




17/42



Surjective mapping

Bijective mapping

Surjective mappingExamples

Example

The function g : IR

IR defined by g(x) = x2 is not

surjective,There is no real number x such that x2 = 1.The function h : IR [0, ) with the same formula as g issurjective.

Given an arbitrary nonnegative real number y, we cansolve y = x2 to get solutions x = y and x = y.


Basic concepts

I j ti j ti bij ti i

Injective mapping

S j ti i


18/42



Surjective mapping

Bijective mapping

Agenda

1 Basic concepts



Bijective mapping


Composition of maps

Inverse maps


Basic concepts

Injective surjective bijective mappings

Injective mapping

Surjective mapping


19/42



Surjective mapping

Bijective mapping

Bijective mappingDefinition

DefinitionThe mapping f : S T is said to be 1-1 correspondence orbijective f is both 1-1 and onto.


Basic concepts


Injective mapping

Surjective mapping


20/42



Surjective mapping

Bijective mapping

Bijective mappingExamples

Example

Figure: A bijective function


Basic concepts


Injective mapping

Surjective mapping


21/42



Surjective mapping

Bijective mapping

Bijective mappingExample

Example

The function f : IR IR defined by f(x) = 2x + 1 isbijective.

Given an arbitrary real number y, we can solve y = 2x + 1

to get exactly one real solution x = (y 1)/2.The function g : IR IR defined by g(x) = x2 is notbijective, for two essentially different reasons.

g is not injective and g is not surjective either. Either one of

these facts is enough to show that g is not bijective.

The function h : [0, ) [0, ) with the same formula asg is bijective.

Given an arbitrary nonnegative number y, we can solve

y = x2 to get exactly one nonnegative solution x =

y.


Basic concepts


Injective mapping

Surjective mapping


22/42



Surjective mapping

Bijective mapping

Remark

Remark

Suppose S and T be finite sets. We havei) If f : S T is injective then N(S) N(T)

ii) If f : S T is surjective then N(S) N(T)iii) If f : S T is bijective then N(S) = N(T)


Basic conceptsInjective, surjective, bijective mappings

Composition of maps

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


23/42


Composition of maps, inverse mapsInverse maps

Agenda

1 Basic concepts



Bijective mapping


Composition of mapsInverse maps



Composition of maps

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


24/42

j , j , j pp g


Composition of mapsIntroduction

Now that we have the notion of a mapping and have singled out

various type of mappings, we might very well ask : "Good and

well. But what can we do with them?".

Consider the situation

g : S T and f : T U.

Given an element s S, then g sends it into the element g(s)in T; so g(s) is ripe for being acted on by f.Thus we get an element f(g(s)) in U. We claim that thisprocedure provides us with a mapping from S to U (Verify!).



Composition of maps

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


25/42

j , j , j pp g


Composition of mapsDefinition

Definition

If g : S T and f : T U, then the composition (or product),denoted by f g, is the mapping f g : S U defined by(f g)(s) = f(g(s)) for every s S.



Composition of maps

Inverse maps


26/42


Composition of mapsExamples

Example

Let mappings f and g be defined by

f(x) = sin x (x IR), g(x) = x2 (x IR)

Then we may form f g and also g f :

IRf

IR

g

IR

IRg IR f IR



Composition of maps

Inverse maps


27/42


Composition of mapsExamples

Example (continue...)

Now

(g f)(x) = g(f(x)) = g(sin x) = (sin x)2 = sin2 x

(f g)(x) = f(g(x)) = f(x2) = sin x2

We remark that f

g and g

f have the same domain but are

different mappings since it is not true that sin2 x = sin x2 for allx IR.



Composition of maps

Inverse maps


28/42


Composition of mapsProperties

Theorem

Let S, T and U be sets and let g : S T and f : T U bemappings.i) If f and g are surjective then f g is surjective

ii) If f and g are injective then f g is injective

iii) If f and g are bijective then f g is bijective



C i i f i

Composition of maps

Inverse maps


29/42



Proof.

We prove i) and leave ii) to be proved in Exercise. i) and ii)

imply iii).Let u U. Then since f is surjective we have t T such thatf(t) = u. Since g is surjective we have s S such that f(s) = t.Then

(f

g)(s) = f(g(s)) = f(t) = u

and so f g is surjective.



C iti f i

Composition of maps

Inverse maps


30/42

Composition of maps, inverse mapsp


Lemma

If f : S T and iT is the identity mapping of T onto itself and iSis the identity mapping of S onto itself, then iT f = f andf iS = f .



Composition of maps inverse maps

Composition of maps

Inverse maps


31/42

Composition of maps, inverse mapsp

Composition of maps

Consider the case of four sets S, T, U, V and three mappingsf, g, h where f : S T, g : T U and h : U V.We may construct the mapping g f and h g and then thefurther compositions h (g f) and (h g) f

The h (g f) and (h g) f are both mappings from S V butthey have been constructed differently. Could they be equal?




Composition of maps

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


32/42


Composition of maps

Example

Let f, g, h be the mappings given by

f(x) = x

2

+ 1 (x ZZ),g(n) =

2

3n (n IN),

h(t) =

t2 + 1 (t OQ),

and so,

ZZf IN g OQ h IR




Composition of maps

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


33/42


Composition of maps


(g f)(x) = g(f(x)) = g(x2 + 1) = 23

(x2 + 1) (x ZZ)

[h (g f)](x) = h( 23

(x2 + 1)) =

(

2

3(x2 + 1))2 + 1

(h

g)(n) = h(g(n)) = h(

2

3

n) = ( 23

n)2 + 1 (n

IN)

[(hg)f](x) = (hg)(f(x)) = (hg)(x2+1) =

(2

3(x2 + 1))2 + 1




Composition of maps

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


34/42


Composition of maps


Thus

[h (g f)](x) = [(h g) f](x) (x ZZ)from which

h (g f) = (h g) f

Lemma

If h : S T , g : T U and h: U V , thenf (g h) = (f g) h.




Composition of maps

Inverse maps


35/42

Co pos t o o aps, e se aps

Proof.

Let a

S, then

[f (g h)](a) = f((g h)(a)) = f(g(h(a)))

and

[(f

g)

h](a) = (f

g)(h(a)) = f(g(h(a))).

Thus

[f (g h)](a) = [(f g) h](a)and since a is arbitrary

f (g h) = (f g) h

There is really no need for parentheses, so we write f (g h)as f

g

h.




Composition of maps

Inverse maps


36/42

p p , p

Agenda

1 Basic concepts



Bijective mapping


Composition of mapsInverse maps




Composition of maps

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


37/42

Inverse mapsIntroduction

Given a mapping f : S T, the f1 does not in general definesa mapping from T to S for several reasons.

If f is not onto, so there is some t in T which is not the

image of any element s in S, so f

1(t) = .If f is not 1-1, then for some t T there are at least twodistinct s1 = s2 in S such that f(s1) = t = f(s2). So f1(t)is not a unique element of S (something we require in our

definition of a mapping).

However, if f is both onto T and 1-1, i.e. f is bijective, then f1

indeed defines a mapping of T onto S.




Composition of maps

Inverse maps


38/42

Inverse maps

Theorem

Bijective mapping f : S T defines a mapping from T onto S.This mapping is called the inverse mapping of f and denoted by

f1.

Example

a) f : IR

IR, y = f(x) = 2x + 1 then x = f1(y) =

y 1

2b) f : IR (0, +), y = f(x) = ex then x = f1(y) = ln y




Composition of maps

Inverse maps


39/42

Inverse mapsProperties

Lemma

If f : S T is a bijection, then(f1)1 = f and f f1 = iT,f1

f

=iS

, where iS

and iT

are the identity mappings of S and

T , respectively.

Lemma (Inverse of composition)

Let f : S

T and g : T

U be bijective mappings. Then

(g f)1 = f1 g1




Composition of maps

Inverse maps


40/42

An interesting subject (?!)

Mappings may appear to be necessary but somewhat

humdrum objects of study(!).

However particular mapping give rise to curious problems.

Consider the mapping f : IN IN given by

f(n) =

12

n (n even)12 (3n+ 1) (n odd)

Thus f(26) = 13, f(25) = 12

(75 + 1) = 38.




Composition of maps

Inverse maps


41/42

An interesting subject (?!)

Suppose we iterate this mapping several times and, by way of

illustration, we follow the effect of the iterations on the number

10. Then

f(10) = 5,

After five iterations on 10 the number 1 appears.




Composition of maps

Inverse maps


42/42

The reader may care to verify that commencing with 65

there are 19 iterations before 1 first appears.

Other integers may be tired at random as test cases, but

the reader is advised to cultivate patience as the number ofiterations before 1 first appears may be quite large.

The conjecture, that for any n IN and for sufficiently manyiterations the number 1 always appears, remain to be proved or

disproved.


Documents

Chapter III - Mapping I