# pigeonhole principle,cardinality,countability

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It describes three phenomenal concepts of discrete mathematics.. Presented in 3rd semester :)TRANSCRIPT

- 1. Pigeonhole Principle, Countability, Cardinality

Presented by:

08-SE-59

08-SE-72

2. What is a Pigeonhole Principle?

3. 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.

4. 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!

5. Pigeons Pigeon holes

6. Daily Life Examples

15 tourists tried to

hike the Washington

mountain. The oldest

of themis 33, while

the youngest one is

20. Then there must beat least 2 tourists of the

same age.

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

same day.

8. 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.

9. 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.

10. Problem Statement

Show that if any 5 no from 1 to 8 are chosen then two of them will add to 9.

11. 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}

12. Each of the 5 nos 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.

13. Cardinality

What do you mean by cardinality?

14. 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!

15. 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.

16. Properties of Cardinality

Reflexive 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

17. 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.

18. Explanation & Example

LetA={(x+1)3| xW & 1 (x+1)3 3000}

Lets find the cardinality of A. After a few calculation we observe that

(0+1)3=1

(1+1)3=8

.

.

(13+1)3=2744

So we have a bijection f={0,1,..,13}->A where

f(x)=(x+1)3.Therefore,|A|={0,1,.,13}=A

19. Cardinality of an Infinite Set

Infinite Set:

A set is said to be infinite if it is equivalent to its proper subset.

20. Explanation & Example

Let S={ n Z+=: n=k2, for some positive integer k}

And Z+ denote the set of positive integers

Let 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+)

k12=k22

k1= k2( but k1 and k2 are positive)

Hence k1=k2

21. f is onto

Suppose n 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.

22. Countability

What does it

mean to say that

a set is

countable?

23. Countable Set

A 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.

24. 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.

25. Techniques to Show Countability

An interesting and useful fact about countability is that

the set NxN is countable.

Theorem:

NxN is a countable set

26. Proof

Note that NxN is countable as a consequence of the definition because the function f: NxN->Ngiven 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 NxNto the set AxB. This result generalizes to the Cartesian product of any finite collection of countable sets.

27. Conclusion

Pigeonhole Principle says that there cant

exist 1-1 correspondence between two

sets those have different cardinality. But we saw that two sets which have 1-1 correspondence have the same cardinality.

28. Thanks...

All Is Well..

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