pigeonhole principle,cardinality,countability

Pigeonhole Principle, Countability, Cardinality

Presented by:08-SE-5908-SE-72

What is a Pigeonhole Principle…?

The Pigeonhole Principle Suppose a flock of pigeons fly into a set of pigeonholes. If there are more pigeons than pigeonholes, then there must be at least 1 pigeonhole that has more than one pigeon in it.

If n items are put into m pigeonholes with n > m, then at least one pigeonhole must contain more than one item.

The Pigeonhole Principle A function from one finite set to a smaller finite set cannot be one-to-one. There must be at least two elements in the domain that have the same image in the co domain…!

Pigeons Pigeon holes

Daily Life Examples 15 tourists tried to hike the

Washington mountain. The

oldest of them is 33,

while the youngest one is 20. Then there

must be at least 2 tourists of the

same age.

There are 380 students at Magic school. There must at least two students whose birthdays happen on a

same day.

65 students appeared in an exam. The possible grades are:

A, B, C and D. There are at least two of them who managed to get the same grade in the exam.

Generalized Pigeonhole Principle If N objects are placed into k boxes,

then there is at least one box containing at least N/k objects…!

For Example:

Among any 100 people there must be at least 100/12 = 9 who were born in the same month.

Problem Statement Show that if any 5

no from 1 to 8 are chosen then two of them will add to 9.

How to Solve…?

Construct 4 different sets each containing two numbers that add up to 9.

A1={1,8}A2={2,7}A3={4,5}A4={3,6}

Each of the 5 no‘s chosen must belong to one of these sets, since there are only 4 sets the pigeon hole principle states that two of the chosen numbers belong to the same sets.

These numbers add up to 9.

CardinalityWhat do you mean by cardinality…?

Cardinality of two Sets Two sets A and B have the same

cardinality if and only if there is a one-to-one correspondence from A to B i.e. there is a function from A to B that is one-to-one and onto…!

How Cardinality is defined...?There are two approaches to cardinality – one which compares sets directly using

bijections and injections and another which uses cardinal numbers. Cardinal Numbers:

No. of elements in a set is called cardinal number.

Properties of CardinalityReflexive property of cardinality; A

has the same cardinality as A.Symmetric property of cardinality; If A

has the same cardinality as B than B has the same cardinality as A.

Transitive property of cardinality; If A has the same cardinality as B and B has the same cardinality as C then A has the same cardinality as C…

Cardinality of a Finite Set

Finite set: A set is called finite if and only if it is

empty set or there is a one-to-one correspondence from

{1,2,_ _,n} to it, where n is a positive integer.

Explanation & ExampleLet A={(x+1)3 | xЄW & 1 ≤ (x+1)3 ≤ 3000}Let’s find the cardinality of A. After a few

calculation we observe that (0+1)3=1 (1+1)3=8

.

. (13+1)3=2744So we have a bijection f={0,1,…..,13}→A wheref(x)=(x+1)3.Therefore,|A|={0,1,….,13}=A

Cardinality of an Infinite SetInfinite Set: A set is said to be infinite if

it is equivalent to its proper subset.

Explanation & ExampleLet S={ n Є Z+=: n=k2, for some positive integer

k}And Z+ denote the set of positive integersLet there be a function P: Z+ S for all positive

Integers. f(k)=k2

f is one-one f(k1)=f(k2) (for all k1, k2

belongs Z+) k1

2=k22

k1= ±k2 ( but k1 and k2 are positive)

Hence k1=k2

f is onto Suppose n Є S by definition of S , n= k2

for some positive integer k. By definition of f

n=f(k).

We can see that there is a one-one correspondence from Z+ to S

So we can say S and Z+ have same cardinality.

CountabilityWhat does it mean to say

that a set iscountable…?

Countable SetA set A is said to be countable if it is either finite

OR its Denumerable Denumerable: If a set is equivalent to the

set of natural numbers N then it is called denumerable set.

Countable Properties

Every subset of N is countable.S is countable if and only if |S| ≤ |N|.Any subset of a countable set is countable.Any image of a countable set is countable.

Techniques to Show CountabilityAn interesting and useful fact about

countability is that the set NxN is countable.

Theorem: NxN is a countable set

Proof Note that NxN is countable as a consequence

of the definition because the function f: NxN→N given by f (m,n) = 2m3n is injective.

This follows because if A and B are countable there are surjections f:N→A and g:N→B. So

fxg : NxN→AxB is a surjection from the countable set NxN

to the set AxB. This result generalizes to the Cartesian product of any finite collection of countable sets.

Conclusion Pigeonhole Principle says that there can’t exist 1-1 correspondence between twosets those have different cardinality. But we

saw that two sets which have 1-1 correspondence have the same cardinality.

Thanks...All Is

Well..