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It describes three phenomenal concepts of discrete mathematics..

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- 1. Pigeonhole Principle, Countability, Cardinality<br />Presented by:<br />08-SE-59<br />08-SE-72<br />
- 2. What is a Pigeonhole Principle…?<br />
- 3. The Pigeonhole Principle<br /> 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.<br /> If n items are put into m pigeonholes with <br />n > m, then at least one pigeonhole must contain more than one item.<br />
- 4. The Pigeonhole Principle<br /> 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…!<br />
- 5. Pigeons Pigeon holes<br />
- 6. Daily Life Examples<br /> 15 tourists tried to <br /> hike the Washington<br /> mountain. The oldest<br /> of them is 33, while<br /> the youngest one is<br /> 20. Then there must be at least 2 tourists of the<br /> same age. <br />
- 7. There are 380 students at Magic school. There must at least two students whose birthdays happen on a<br /> same day.<br />
- 8. 65 students appeared in an exam. The possible grades are:<br /> A, B, C and D. There are at least two of them who managed to get the same grade in the exam.<br />
- 9. Generalized Pigeonhole Principle<br /> If N objects are placed into k boxes, then there is at least one box containing at least N/k objects…!<br />For Example:<br /> Among any 100 people there must be at least 100/12 = 9 who were born in the same month.<br />
- 10. Problem Statement<br /> Show that if any 5 no from 1 to 8 are chosen then two of them will add to 9.<br />
- 11. How to Solve…?<br />Construct 4 different sets each containing two numbers that add up to 9.<br />A1={1,8}<br />A2={2,7}<br />A3={4,5}<br />A4={3,6}<br />
- 12. 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.<br />These numbers add up to 9.<br />
- 13. Cardinality<br />What do you mean by cardinality…?<br />
- 14. Cardinality of two Sets<br />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…!<br />
- 15. How Cardinality is defined...?<br />There are two approaches to cardinality – <br />one which compares sets directly using bijections and injections<br /> and another which uses cardinal numbers. <br />Cardinal Numbers:<br />No. of elements in a set is called cardinal number.<br />
- 16. Properties of Cardinality<br />Reflexive property of cardinality; A has the same cardinality as A.<br />Symmetric property of cardinality; If A has the same cardinality as B than B has the same cardinality as A.<br />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…<br />
- 17. Cardinality of a Finite Set <br />Finite set:<br /> A set is called finite if and only if it is empty set or there is a one-to-one correspondence from <br /> {1,2,_ _,n} to it, where n is a positive integer.<br />
- 18. Explanation & Example<br />Let A={(x+1)3| xЄW & 1 ≤ (x+1)3 ≤ 3000}<br />Let’s find the cardinality of A. After a few calculation we observe that<br /> (0+1)3=1<br /> (1+1)3=8<br />.<br /> .<br /> (13+1)3=2744<br />So we have a bijection f={0,1,…..,13}->A where<br />f(x)=(x+1)3.Therefore,|A|={0,1,….,13}=A<br />
- 19. Cardinality of an Infinite Set<br />Infinite Set:<br /> A set is said to be infinite if it is equivalent to its proper subset.<br />
- 20. Explanation & Example<br />Let S={ n Є Z+=: n=k2, for some positive integer k}<br />And Z+ denote the set of positive integers<br />Let there be a function P: Z+ S for all positive Integers.<br /> f(k)=k2<br /> f is one-one <br /> f(k1)=f(k2) (for all k1, k2 belongs Z+)<br />k12=k22<br /> k1= ±k2 ( but k1 and k2 are positive)<br /> Hence k1=k2<br />
- 21. f is onto<br /> Suppose n Є S by definition of S , n= k2 for some positive integer k.<br /> By definition of f <br /> n=f(k).<br /> We can see that there is a one-one correspondence from Z+ to S<br /> So we can say S and Z+ have same cardinality.<br />
- 22. Countability<br />What does it<br /> mean to say that<br /> a set is<br />countable…? <br />
- 23. Countable Set<br />A set A is said to be countable if<br /> it is either finite <br />OR <br />its Denumerable <br /> Denumerable: If a set is equivalent to the set of natural numbers N then it is called denumerable set.<br />
- 24. Countable Properties<br />Every subset of N is countable.<br />S is countable if and only if |S| ≤ |N|.<br />Any subset of a countable set is countable.<br />Any image of a countable set is countable.<br />
- 25. Techniques to Show Countability<br />An interesting and useful fact about countability is that<br /> the set NxN is countable.<br />Theorem:<br />NxN is a countable set<br />
- 26. Proof <br /> Note that NxN is countable as a consequence of the definition because the function f: NxN->Ngiven by f (m,n) = 2m3n is injective. <br /> This follows because if A and B are countable there are surjections f:N->A and g:N->B. So<br />fxg : NxN->AxB<br /> 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.<br />
- 27. Conclusion<br /> Pigeonhole Principle says that there can’t<br /> exist 1-1 correspondence between two<br />sets those have different cardinality. But we saw that two sets which have 1-1 correspondence have the same cardinality.<br />
- 28. Thanks...<br />All Is Well..<br />

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