1 2 3,, - indian institute of technology · pdf file• it is very difﬁcult to pinpoint...

December16, 2003

Let us count

1, 2, 3, . . . ,∞

Prof. Inder K. Rana

[email protected]

Department of Mathematics

Indian Institute of Technology Bombay, Powai

Mumbai, 400076, India

Inder K. Rana, IIT Bombay – p. 1/30

December16, 2003

Origin of counting and number symbols

• It is very difficult to pinpoint the origin of numbers and theprocess of counting.


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• Of course, this must have been a process of gradualawareness developing over centuries, rather than a discovery.


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• The need to express these similarities or differences insymbols − first orally and then written − must have given riseto the process of counting and to number symbols.


December16, 2003




• The need to express these similarities or differences insymbols − first orally and then written − must have given riseto the process of counting and to number symbols.

• Anthropologists have discoveries to their credit to claim thatthis process must have begun at least about 30,000 yearsback.


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• Around 1400 A.D, these symbols and the decimal place-valuesystem reached Europe and rest of the world via the Arabtraders.


December16, 2003

Number symbols

Evolution of number symbols


December16, 2003

Number symbols

• The current number symbols 1, 2, . . . , 9 and the decimalplace value system help us to represent all the numbers.


December16, 2003

Number symbols


• These numbers are called natural numbers and we denotethe set of natural numbers by IN.


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Number symbols



• As we know, every method of counting is a process ofassociating, with each object to be counted, a unique naturalnumber in a one-to-one manner.


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Number symbols




• The process of counting is based on the following:

Pigeon hole principle:


December16, 2003

Number symbols






1. If m pigeons are put in m holes then there is an empty holeif and only if there is a hole with more than one pigeon.


December16, 2003

Number symbols






1. If m pigeons are put in m holes then there is an empty holeif and only if there is a hole with more than one pigeon.

2. If m pigeons are put in n holes, m > n, then there is a holewith more than one pigeon.


December16, 2003

Pigeon Hole Principle

• Mathematically these can be stated as follows:


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• Mathematically these can be stated as follows:1. For n ∈ IN, every one-one map f : {1, 2, . . . , n} → {1, 2, . . . , n} is

also onto.


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also onto.

2. For n,m ∈ IN there exists a bijective mapΦ : {1, 2, . . . , n} → {1, 2, . . . ,m} if and only if n = m.


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also onto.


• Some applications of the Pigeon Hole Principle are as follows:


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also onto.


• Some applications of the Pigeon Hole Principle are as follows:1. Among any group of 8 persons, two of them have same day of the week

as birthday.


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also onto.


• Some applications of the Pigeon Hole Principle are as follows:1. Among any group of 8 persons, two of them have same day of the week

as birthday.

2. Among any N positive integers, there exists a pair whose difference isdivisible by N − 1.


December16, 2003

Application of PHP

• Proof of (2): Let a1, a2, . . . , aN be the given numbers.


December16, 2003

Application of PHP


For each ai, let ri be the remainder that results from dividingai by N − 1.


December16, 2003

Application of PHP



Each ri can take one the values 0, 1, ..., N − 2.


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Application of PHP




Thus there are N − 1 possible values for N remainders:r1, r2, . . . , rN .


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Application of PHP





Hence, by the pigeon hole principle, there must be two of ther′is that are the same,i,e., rj = rk for some pair j and k.


December16, 2003

Application of PHP





Hence, by the pigeon hole principle, there must be two of ther′is that are the same,i,e., rj = rk for some pair j and k.

But then, aj and ak have the same remainder when divided byN − 1, and so their difference aj − ak is divisible by N − 1.


December16, 2003

Finite sets

In view of the pigeon hole principle, we have the following:


December16, 2003

Finite sets

In view of the pigeon hole principle, we have the following:• Definition:

A set A is said to be countably finite if either A = ∅ or thereexists an n ∈ IN and a bijective map φ : A→ {1, 2, . . . , n}.


December16, 2003

Finite sets



Note that by the PHP, such a n is unique, and we say that Ahas n elements.


December16, 2003

Finite sets




The following properties of finite sets are easy to check:


December16, 2003

Finite sets




The following properties of finite sets are easy to check:• Theorem:

Let A and B be two sets. Then the following hold:


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Finite sets






(i) If A is countably finite and B is countably finite, so also is A ∪B.


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Finite sets






(i) If A is countably finite and B is countably finite, so also is A ∪B.

(ii) If A1, A2, . . . , An are finite sets, Then so is the set ∪ni=1Ai.


December16, 2003

Countably-infinite

At this point it is natural to ask the question:

How to analyze sets which are not finite?


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Countably-infinite



For example, IN, the set of all natural numbers is not finite!


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Countably-infinite




It is natural to call such sets as infinite sets.


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Countably-infinite





The concepts of ‘infinite’ and ‘infinite set’ eludedmathematicians and philosophers over the centuries.


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Countably-infinite






• The Greek mathematician, Pythagoras (∼ 585–500 B.C.),associated good and evil with the limited and the unlimited,respectively.


December16, 2003

Countably-infinite






• The Greek mathematician, Pythagoras (∼ 585–500 B.C.),associated good and evil with the limited and the unlimited,respectively.

• “The infinite is imperfect, unfinished and therefore,unthinkable; it is formless and confused," said, Aristotle(384–322 B.C.).


December16, 2003

Infinity

• The Roman Emperor and philosopher Marcus Aqarchus(121-180 A.D.) said, “Infinity is a fathomless gulf, into which allthings vanish".


December16, 2003

Infinity


• English philosopher Thomas Hobbes (1588–1679) said,“When we say any thing is infinite, we signify only that we arenot able conceive the ends and bounds of the thing named".


December16, 2003

Infinity



• The German mathematician Carl Friedrich Gauss(1777–1855) said, “Infinity is only a figure of speech, meaninga limit to which certain ratios may approach as closely asdesired, when others are permitted to increase indefinitely".


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Infinity




• Infinity is indeed strange and abstract.


December16, 2003

Infinity




• Infinity is indeed strange and abstract.• Let us look at some examples.


December16, 2003

Infinity




• Infinity is indeed strange and abstract.• Let us look at some examples.• Zeno’s Paradox:


December16, 2003

Infinity




• Infinity is indeed strange and abstract.• Let us look at some examples.• Zeno’s Paradox:

A man standing in a room can not walk to the wall he is facing.


December16, 2003

Zeno’s paradx

◦ ◦ ◦

1/2 1/4 1/8

6

Man → Wall

��


December16, 2003

Zeno’s paradx

◦ ◦ ◦

1/2 1/4 1/8

6

Man → Wall

��

Because, in order to do so, he would have to go half thedistance,

◦ ◦ ◦

1/2 1/4 1/8

6

Man →

Wall

��


December16, 2003

Zeno’s paradx

◦ ◦ ◦

1/2 1/4 1/8

6

Man → Wall

��

Because, in order to do so, he would have to go half thedistance,

◦ ◦ ◦

1/2 1/4 1/8

6

Man →

Wall

��

then half the remaining distance,Inder K. Rana, IIT Bombay – p. 12/30

December16, 2003

Zeno’s paradox

• and then again half of what shall remain.


December16, 2003

Zeno’s paradox


This process can always be continued and it can never end.


December16, 2003

Zeno’s paradox



However, physically the man walks to the wall in finite time.


December16, 2003

Zeno’s paradox



However, physically the man walks to the wall in finite time.• The paradox is that there are infinite stages and how they can

be covered in finite time?


December16, 2003

Zeno’s paradox




be covered in finite time?• Explanation: Let the man walk with a constant speed and takea minutes to cover the first half of the distance.


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Zeno’s paradox





The next half will be covered in a/2 minutes, the half of theremaining half in a/4 minutes, and so on.


December16, 2003

Zeno’s paradox





The next half will be covered in a/2 minutes, the half of theremaining half in a/4 minutes, and so on.

The time consumed at the nth stage will be

Tn = a+a

2+

a

22+ · · · +

a

2n−1,

which is ’convergent’ to a.


December16, 2003

Hotel Hilbert

• Another illustration of the mysteries of the infinite is thefollowing :


December16, 2003

Hotel Hilbert


• Problem:To construct a hotel which always has vacant rooms!


December16, 2003

Hotel Hilbert



Solution:Construct a hotel with infinite number of rooms.


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Hotel Hilbert




The solution was implemented and the new hotel was called:


December16, 2003

Hotel Hilbert





... HHHHHHHHHOTEL HILBERT TTTTTTTT ...


December16, 2003

Hotel Hilbert






The rooms in the hotel were along one big long corridor,


December16, 2003

Hotel Hilbert







with all the odd numbered rooms 1, 3, 5, 7, . . . , on the left handside of the hall and at the right side were all the evennumbered rooms 2, 4, 6, 8, . . ..


December16, 2003

Hotel Hilbert







with all the odd numbered rooms 1, 3, 5, 7, . . . , on the left handside of the hall and at the right side were all the evennumbered rooms 2, 4, 6, 8, . . ..

Everybody was happy that this hotel had accommodation forevery visitor.


December16, 2003

Hotel Hilbert

However, one day the most unbelievable thing happened anda problem came up:


December16, 2003

Hotel Hilbert


• New Problem:One day the receptionist wanted to accommodate a newvisitor, but observed that the hotel was full!!


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Hotel Hilbert



• The perplexed receptionist went to the manager of the hotel.


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Hotel Hilbert



• The perplexed receptionist went to the manager of the hotel.• The manager thought for a while, and solved the problem as

follows:


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Hotel Hilbert




follows:• Solution: Move occupant of Room1 to Room2, move occupant

of Room2 to Room3, and in general


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Hotel Hilbert






move occupant of RoomN to RoomN + 1.


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Hotel Hilbert







This will make the Room1 available for the new guest.


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Hotel Hilbert







This will make the Room1 available for the new guest.

Business went smoothly for the hotel till one day, thereceptionist came back to the manager with yet anotherproblem:


December16, 2003

Hotel Hilbert

• Yet another problem:Infinite number of new visitors had arrived asking foraccommodation, and the hotel was full !!!


December16, 2003

Hotel Hilbert


The manger was out of his wits to solve the problem. Luckily,a mathematician friend of his was in his office, and heproposed a solution the manager.


December16, 2003

Hotel Hilbert



Solution:Move occupant of Room1 to Room2, move occupant ofRoom2 to Room4, and in general


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Hotel Hilbert




move occupant of RoomN to Room2N.


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Hotel Hilbert





This will shift all the present occupants of the hotel on theright-hand side of the hotel, i.e., in the even-numbered rooms,making all the odd numbered rooms vacant.


December16, 2003

Hotel Hilbert





This will shift all the present occupants of the hotel on theright-hand side of the hotel, i.e., in the even-numbered rooms,making all the odd numbered rooms vacant.

Accommodate all the new guests (infinite of them) into all theodd-numbered (vacant) rooms.


December16, 2003

Hotel Hilbert

Though happy for the time being, the problems for the hotelmanager did not end there. Another problem arose whosesolution seemed impossible.


December16, 2003

Hotel Hilbert


• Super problem:Accommodation had to be arranged for the participants ofYouth Festival teams of infinite number of colleges:C1, C2, C3, . . . , each college team consisting of infinitenumber of participants, and the hotel was full !!!!


December16, 2003

Hotel Hilbert



Hotel manager was fully shaken by this crises. He rushed tohis mathematician friend for help. This time themathematician friend had to scratched his head for sometime, but he did help his friend.


December16, 2003

Hotel Hilbert




• The solution is based on the fact:there are infinite number of primes,


December16, 2003

Hotel Hilbert




• The solution is based on the fact:there are infinite number of primes,

and the solution is as follows:


December16, 2003

Countably-infinite

• Solution:Let the primes be denoted byp0 = 2, p1 = 3, p2 = 5, p3 = 7, . . . .


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Countably-infinite


First step of the solution is to shift the current occupants ofRoomN to Room2N .


December16, 2003

Countably-infinite



• Note that the current occupants will get shifted to the evennumbered rooms, namely Rooms2, 4, 8, 16, 32, 64, . . . , leavingall other rooms vacant.


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Countably-infinite




Next, put the jth member of the Cith team in room(pi)

j , j = i, 2, 3, . . . .


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Countably-infinite





j , j = i, 2, 3, . . . .

• Note that (pi)j is an odd number for every j.


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Countably-infinite





j , j = i, 2, 3, . . . .


This solution not only accommodated all the new visitorswithout any clashes, in fact


December16, 2003

Countably-infinite





j , j = i, 2, 3, . . . .


This solution not only accommodated all the new visitorswithout any clashes, in fact

infinite number of rooms, for example Rooms6, 12, 18, . . . orRooms10, 20, 30, . . . , remained vacant.


December16, 2003

Countably-infinite

• Thus, even though initially no room was vacant, hotel was ableto accommodate infinite number of teams, each having infinitemembers, and still leaving infinite number of rooms vacant!


December16, 2003

Countably-infinite


The solutions of the above problems are in fact theorems inset theory. Let us first make the following:


December16, 2003

Countably-infinite



• Definition:


December16, 2003

Countably-infinite



• Definition:

1. A nonempty set B is said to be countably infinite if thereexists a bijective map ψ : B → IN.


December16, 2003

Countably-infinite



• Definition:


2. A set C is said to be countable if either it is countably finiteor countably infinite.


December16, 2003

Countably-infinite



• Definition:



• Theorem:


December16, 2003

Countably-infinite



• Definition:



• Theorem:

(i) If A is countably finite and B is countably infinite, A ∪B is countablyinfinite.


December16, 2003

Countably-infinite



• Definition:



• Theorem:

(i) If A is countably finite and B is countably infinite, A ∪B is countablyinfinite.

(ii) If A and B are countably infinite, so also is A ∪B.


December16, 2003

Countably-infinite

(iii) If An is a countable set for each n ∈ IN, then A :=⋃

∞

n=1An is also acountable set.


December16, 2003

Countably-infinite


∞


• Examples:


December16, 2003

Countably-infinite


∞


• Examples:

1. The set IN itself is countably infinite.


December16, 2003

Countably-infinite


∞


• Examples:

1. The set IN itself is countably infinite.2. INe, the set of all even natural numbers and INo, the set of

all odd natural numbers are both countably infinite.


December16, 2003

Countably-infinite


∞


• Examples:


all odd natural numbers are both countably infinite.3. The set IN × IN is countably infinite since g : IN × IN → IN

defined byg(m,n) := 2m−1(2n− 1),

is a bijective map.


December16, 2003

Countably-infinite


∞


• Examples:




is a bijective map.4. The sets ZZ and IQ are countably infinite.


December16, 2003

Countably-infinite


∞


• Examples:




is a bijective map.4. The sets ZZ and IQ are countably infinite.5. The set of all algebraic numbers is countably infinite.


December16, 2003

Cantor’s Theorem

• Question:Is P(IN), the set of all subsets of IN, countably infinite?


December16, 2003

Cantor’s Theorem


To answer this we need the following:


December16, 2003

Cantor’s Theorem


To answer this we need the following:• Theorem (Cantor):

Let X be any nonempty set. Then there exists a one-one mapping of Xinto P(X) but there exists no bijection between X and P(X).


December16, 2003

Cantor’s Theorem




• Click here to see the proof.


December16, 2003

Cantor’s Theorem





As a consequence of this, we have:


December16, 2003

Cantor’s Theorem





As a consequence of this, we have:• Corollary:

If X is a countably infinite set, then P(X) is not countably infinite.


December16, 2003

Cantor’s Theorem







• Thus, in some sense P(IN) has much more number ofelements than that of IN.


December16, 2003

Cantor’s Theorem







• Thus, in some sense P(IN) has much more number ofelements than that of IN.

Let us give a name to the ’number’ of elements of a set.


December16, 2003

Cardinality

• Definition:Let X be any nonempty set.


December16, 2003

Cardinality


(i) If X is a finite set, we define its cardinality to be the numberof elements in X,and is denoted it by that natural number.


December16, 2003

Cardinality



(ii) If X is countably infinite, we say X has cardinalityaleph-nought and denote it by ℵ0.


December16, 2003

Cardinality




The question arises:What is the cardinality of a set which is not countable?


December16, 2003

Cardinality





To answer this, let us note that if X is a finite set with melements, then P(X) is also finite with 2m elements.Thus wemake the following:


December16, 2003

Cardinality





To answer this, let us note that if X is a finite set with melements, then P(X) is also finite with 2m elements.Thus wemake the following:

• Definition:For a set X whose cardinality is defined and is card(X), let usdenote the cardinality of P(X) by 2card(X).


December16, 2003

Infinity of Infinities

Thus,

card(IN) = ℵ0, card(P(IN)) = 2ℵ0 , card(P(P(IN))) = 22ℵ

0 ,

and so on.


December16, 2003


Thus,


0 ,

and so on.

Note that, ℵ0, 2ℵ0 , 22ℵ

0 , . . . , are all infinities.


December16, 2003


Thus,


0 ,

and so on.


0 , . . . , are all infinities.• Thus, we have infinity of infinities: with

1 < 2 < 3 < · · · < ℵ0 < 2ℵ0 := c < 2c < 22c

< · · · .


December16, 2003


Thus,


0 ,

and so on.



1 < 2 < 3 < · · · < ℵ0 < 2ℵ0 := c < 2c < 22c

< · · · .

• Thus, there is counting beyond infinity.


December16, 2003


Thus,


0 ,

and so on.



1 < 2 < 3 < · · · < ℵ0 < 2ℵ0 := c < 2c < 22c

< · · · .

• Thus, there is counting beyond infinity.• The above process of counting enables us to define the

cardinality (the number of elements ) for a set which is eitherfinite, or countably-infinite, or for the power set of a set, forwhich the notion of cardinality has already been defined.


December16, 2003

Equivalance of sets

• This raises the following natural questions:


December16, 2003

Equivalance of sets

• This raises the following natural questions:• How do we compare the number of elements for given sets?


December16, 2003

Equivalance of sets

• This raises the following natural questions:• How do we compare the number of elements for given sets?• How to define the cardinality of any given set which is not of

the type as described above?


December16, 2003

Equivalance of sets



Assuming that one already has the notion of cardinalitydefined for every set, it is easy to answer the secondquestion.


December16, 2003

Equivalance of sets




Definition:Given two sets A and B, we say A and B have the samecardinality if there exists a one-one, onto map f : A→ B. Wewrite this as A ∼ B.


December16, 2003

Equivalance of sets




Definition:Given two sets A and B, we say A and B have the samecardinality if there exists a one-one, onto map f : A→ B. Wewrite this as A ∼ B.

• On any given collection S of sets, the relation ∼ is anequivalence relation.


December16, 2003

CBS-theorem

• Examples:


December16, 2003

CBS-theorem

• Examples:

1. For n ∈ IN, {1, 2, . . . , n} ∼ {1, 2, . . . ,m} iff n = m.


December16, 2003

CBS-theorem

• Examples:

1. For n ∈ IN, {1, 2, . . . , n} ∼ {1, 2, . . . ,m} iff n = m.

2. The set IN is not equivalent to [1, n] for any n ∈ IN.


December16, 2003

CBS-theorem

• Examples:

1. For n ∈ IN, {1, 2, . . . , n} ∼ {1, 2, . . . ,m} iff n = m.


3. Clearly any two of the sets IN, ZZ+, ZZ−, IQ and IN × IN areequivalent to one another.


December16, 2003

CBS-theorem

• Examples:

1. For n ∈ IN, {1, 2, . . . , n} ∼ {1, 2, . . . ,m} iff n = m.



4. For a ∈ IR, a > 0, (0, 1) ∼ (0, a).


December16, 2003

CBS-theorem

• Examples:

1. For n ∈ IN, {1, 2, . . . , n} ∼ {1, 2, . . . ,m} iff n = m.



4. For a ∈ IR, a > 0, (0, 1) ∼ (0, a).

5. (0, 1) ∼ IR, since x 7−→x

|x| + 1from IR to (0, 1) is a one-one,

onto map.


December16, 2003

CBS-theorem

• Examples:

1. For n ∈ IN, {1, 2, . . . , n} ∼ {1, 2, . . . ,m} iff n = m.



4. For a ∈ IR, a > 0, (0, 1) ∼ (0, a).

5. (0, 1) ∼ IR, since x 7−→x


onto map.

• An important theorem which allows one to check theequivalence of two sets is the following:Theorem (Cantor-Bernstein-Schr oder):If X and Y are two nonempty sets such that each is equivalent to a subsetof the other then they are equivalent.


December16, 2003

2ℵ0 = c

Examples:


December16, 2003

2ℵ0 = c

Examples:

1. Let X be any subset of IR such that X includes a nonemptyopen interval. Then X ∼ IR.


December16, 2003

2ℵ0 = c

Examples:


To see this, let (a, b) ⊆ X. Since X ∼ X ⊆ IR andIR ∼ (0, 1) ∼⊆ X, by CBS- theorem, X ∼ IR.


December16, 2003

2ℵ0 = c

Examples:



2. Any two intervals in IR are equivalent.


December16, 2003

2ℵ0 = c

Examples:



2. Any two intervals in IR are equivalent.3. P(IN) ∼ I := (0, 1) ∼ IR.


December16, 2003

2ℵ0 = c

Examples:




To see this let A ∈ P(IN) and let p(A) ∈ I be the point withdecimal expansion p(A) = ·x1x2x3 . . . where xi = 1 if i ∈ Aand xi = 0 if i 6∈ A.


December16, 2003

2ℵ0 = c

Examples:





Then p is a one-one function from P(IN) to I.


December16, 2003

2ℵ0 = c

Examples:





Then p is a one-one function from P(IN) to I.Also for any x ∈ (0, 1), let x = ·a1a2a3 . . . be its decimalexpansion. Let s(x) := {n ∈ IN|an = 1}.


December16, 2003

2ℵ0 = c

Examples:






Then s(x) ∈ P(IN) and clearly the map x 7−→ s(x) is aone-one function from I into P(X).


December16, 2003

2ℵ0 = c

Examples:






Then s(x) ∈ P(IN) and clearly the map x 7−→ s(x) is aone-one function from I into P(X).

Hence, by the CBS-theorem I ∼ P(IN).Inder K. Rana, IIT Bombay – p. 26/30

December16, 2003

Dedekind-infinite

• The notion of ’cardinality of a set’ or of ’cardinal number’requires more elaborate discussion, see

"Naive Set Theory by P. R. Halmos.


December16, 2003

Dedekind-infinite



• In our discussions, defined countably-finite andcountably-infinite in comparison with the set of naturalnumbers.


December16, 2003

Dedekind-infinite




An ’intrinsic definition’ of a set to be infinite, which does notmake use of natural numbers, was given by RichardDedekind:


December16, 2003

Dedekind-infinite




An ’intrinsic definition’ of a set to be infinite, which does notmake use of natural numbers, was given by RichardDedekind:

• Definition:A set is called an infinite set if it can be put in one-to-onecorrespondence with a proper subset of itself. And a set iscalled finite if it is not infinite.


December16, 2003

Dedekind-infinite

• Examples:


December16, 2003

Dedekind-infinite

• Examples:

1. IN is infinite as it is in one-to-one correspondence with theproper subset INo.


December16, 2003

Dedekind-infinite

• Examples:


2. Each of , ZZ , IQ, and IR is an infinite set.


December16, 2003

Dedekind-infinite

• Examples:


2. Each of , ZZ , IQ, and IR is an infinite set.3. For every n ∈ IN, the set {1, 2, . . . , n} is finite.


December16, 2003

Dedekind-Infinite

• "The Whole is there, The Whole is here,


December16, 2003

Dedekind-Infinite


From the Whole emanates the Whole,


December16, 2003

Dedekind-Infinite



Taking away the Whole from the Whole,


December16, 2003

Dedekind-Infinite



Taking away the Whole from the Whole,

What remains is the Whole."


December16, 2003

Set Theory

Thus we have two ways of saying a set is finite or not.


December16, 2003

Set Theory


The natural question arises:

Are these two definitions equivalent?


December16, 2003

Set Theory



Are these two definitions equivalent?• Answer to this question leads to some fundamental concepts

and questions in set theory, like:


December16, 2003

Set Theory





What is a set?


December16, 2003

Set Theory





What is a set?

What are natural numbers?


December16, 2003

Set Theory





What is a set?


In case you would like to know more about this, have a look atthe following book:


December16, 2003

Set Theory





What is a set?


In case you would like to know more about this, have a look atthe following book:

FROM NUMBERS TO ANALYSIS

INDER K. RANA

WORLD SCIENTIFIC PRESS, SINGAPORE


1 2 3,, - indian institute of technology · pdf file• it is very difﬁcult to pinpoint...

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