pigeonhole principle,cardinality,countability

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

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  • 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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