Discrete Mathematics

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1 Disrete Mathematis -- Chapter 8: The Priniple i of Inlusion and Exlusion Hung-Yu Kao Department of Computer iene and Information Engineering, ational lcheng Kung University

2 Outline The Priniple of Inlusion and Exlusion Generalization of the Priniple Derangements: othing Is in Its Right Plae Rook Polynomials Arrangements with Forbidden Positions

3 8. The priniple of Inlusion and Exlusion For a given finite set with onditions C i [ ] and or [ ] [ ]

4 Four sets Ex 8. : Four sets ] [ ] [ For eah element x we have five ases: ] [ ] [ For eah element x, we have five ases: 0 x satisfies none of the four onditions; x satisfies only one of the four onditions; x satisfies only one of the four onditions; x satisfies exatly two of the four onditions; x satisfies exatly three of the four onditions; x satisfies exatly three of the four onditions; x satisfies all the four onditions. Disrete Mathematis Disrete Mathematis CH8 CH8 009 pring 009 pring

5 Four sets ] [ ] [ ] [. x satisfies no ondition. x is ounted one in and one in. []. x satisfies. It is not ounted on the left side. It is ounted one in and one in. [0-0]. x satisfies and. It is not ounted on the left side. It is ounted one in,, and.. x satisfies, and. It is not ounted on the left side. It is ounted one in and ] 0 0 [ one in,,,,,, and. 5. x satisfies all onditions. It is not ounted on the left side. It is ounted ] 0 0 [ one in all the subsets on the right side. ] 0 0 [ Disrete Mathematis Disrete Mathematis CH8 CH8 009 pring 009 pring 5

6 Four sets 6

7 The Priniple of Inlusion and Exlusion Theorem 8.:, onditions i, i t denote the number of elements of t that satisfy none of the onditions. i i j i t i< j t i< j< k t i j k t t 7

8 8

9 The Priniple of Inlusion and Exlusion Corollary 8.: or or... or t. ome notation for simplifying Theorem 8. 9

10 The Priniple of Inlusion and Exlusion Ex 8. : Determine the number of positive integers n where n 00 and n is not divisible by, or 5. Condition if n is divisible by. Condition if n is divisible by. Condition i if n is divisible i ibl by 5. Then the answer to this problem is 00 /* 6 [ 0 ] [ 00 [50 0] [6 0 6] 6. ] 0

11 The Priniple of Inlusion and Exlusion Ex 8.5 : Determine the number of nonnegative integer solutions to the equation x x x x 8 and x i 7 for all i. We say that a solution x, x, x, x satisfies ondition i if x i > 7 i.e., x i 8. Then the answer to this problem is

12 The Priniple of Inlusion and Exlusion Ex 8.6 : For finite sets A, B, where A m n B, and funtion f: A B, determine the number of onto funtions f. Let A {a, a,, a m } and B {b, b,, b n }. Let i be the ondition that b i is not in the range of f. Then is the number of funtions in that have bi in their range. Then the answer to this problem is.... n n i 0 n m m n m n n m n m i j n

13 The Priniple of Inlusion and Exlusion Ex 8.7 : In how many ways an the 6 letters of the alphabet be permuted so that none of the patterns ar, dog, pun, or byte ours? Ex 8.8 : Let φn be the number of positive integers m, where m < n and gdm, n, i.e., m and n are relatively prime. e e e e Consider n p p p p For i, let i denote that k is divisible by p i. 0 n; i n/p i ; i j n/p i p j ; Then the answer to this problem is.

14 Disrete Mathematis Disrete Mathematis CH8 CH8 009 pring 009 pring e e e e p p p p p p p p

15 The Priniple of Inlusion and Exlusion Ex 8.9 : ix married ouples are to be seated at a irular table. In how many ways an they arrange themselves so that no wife sits next to her husband? For i 6, let i denote the ondition where a seating arrangement has ouple i seated next to eah other. i -! Then the answer to this problem is

16 The Priniple of Inlusion and Exlusion Ex : In a ertain area of the ountryside are five villages. An engineer is to devise a system of two-way roads so that after the system is ompleted, no village will be isolated. In how many ways an he do this? Let i denote the ondition that a system of these roads isolates village a, b,, d, and e, respetively. C5, C, 6

17 8. Generalizations of the Priniple E m denotes the number of elements in that satisfy exatly m of the t onditions. E t t E t t E - [ ] - E E 8 t t t. t t. 7

18 Generalizations of the Priniple E 6 E 6 E E E Disrete Mathematis Disrete Mathematis CH8 CH8 009 pring 009 pring 8

19 Theorem 8. t m t m m E. t m t t m t m m m m m m E Disrete Mathematis Disrete Mathematis CH8 CH8 009 pring 009 pring 9

20 0

21 Corollary 8. Let L m denotes the number of elements in that satisfy at least m of the t onditions.

22 8. Derangements: othing Is in Its Right Plae x e n x x x x, e!! n! n!! n 0 n 0! e , -/!-/! -/7! n Derangement means that t all numbers are in the wrong positions. Ex 8. : Determine the number of derangements of,,,0. Let i be the ondition that integer i is in the ith plae.

23 Derangements: othing Is in Its Right Plae The general formula: d n n! e n![!!... ]! n! Pd n /n! Ex 8. : Peggy has seven books and hires seven reviewers. he wants two reviewers per book. In how many ways an she make the distributions? The first time: 7! ways The seond time: d 77!* e - Ways different position Totally, we have 7! d 7 ways

24 8. Rook Polynomials In Fig. 8.6, we want to determine the number of ways in whih k rooks an be plaed on the unshaded squares of this hessboard so that no two of them an take eah other that is, no two of them are in the same row or olumn of the hessboard C. This number is denoted as r k C. 5 6

25 Rook Polynomials In Fig. 8.6, we have r 0, r 6, r 8, r and r k 0 for k. Rook polynomial: rc, x 6x8x x. For eah k 0, the oeffiient of x k is the number of ways we an plae k nontaking rooks on hessboard C

26 Disjoint ubboards Break up a larger board into smaller subboards. In Fig. 8.7, the hessboard ontains two disjoint g, j subboards that have no squares in the same olumn or row of C. rc, x rc, x. rc, x C C 0 7,, x x x x C r x x x C r C,, , 5 x C r x C r x x x x x x C r Disrete Mathematis Disrete Mathematis CH8 CH8 009 pring 009 pring 6

27 Disjoint ubboards r for C In general, if C is a hessboard made up of pairwise disjoint subboards C, C,, C n, then rc, x rc, x. rc, x. rc n, x. 7

28 Reursive Formula Fig. 8.8 a, For a given designated square *, we either b plae one rook here, or do not use this square. r k Cr r k- C s r k C e C s : denote the remaining smaller subboard Fig. 8.8b C e: C with the one designed square eliminated Fig. 8.8 r k Cx k r k- C s x k r k C e x k for k n. use not use 8

29 Reursive Formula r 0 C s x 0 typo in p05 9

30 Apply the Reursive Formula use * not use * 0

31 8.5 Arrangements with Forbidden Positions Ex 8.5 : In making seating arrangements, the shaded square of the figure means relative R i will not sit at table T j. Determine the number of ways that we an seat these four relatives at five different tables. Let be the total number of ways we an plae the four relatives. 5! Let i be the ondition that R i is seated in a forbidden position but at different tables.!! R T or R T! R T?!!!!? 7! number of shaded squares R R R R T T T T T 5

32 Arrangements with Forbidden Positions 6!! R T or R T!?,?,?,? Observation: 6 is the number of ways two nontaking rooks an be plaed on the shaded hessboard. Let r i be the number of ways in whih it is possible to plae i nontaking rooks on the shaded hessboard. For all 0 i, i r i5 - i! Deompose C into the disjoint subboards in the upper left and lower right orners. rc,, x x x x x 7x 6x x x 0 5! 7! 6!!! i r 5 i! 5 i 0 i

33 Arrangements with Forbidden Positions Ex 8.6 : We roll two die six times, where one is red ddie and the other green die. We know the following pairs did not our:,,,,, 5,,,,,, 5 and 6, 6. What is the probability that we obtain all six values both on red die and green die? One of solutions is like,,,,,,,, 5, 6, 6, 5. In Fig. 8.0b, hessboard C with seven shaded squares rc, x xx x 7x7x7 9x 0x x 5 i denotes that all six values our on both the red and green dies, but i on the red die is paired with one of the forbidden numbers on the green die.

34 Figure 8.0 a 5 6 b

35 Arrangements with Forbidden Positions 5

36 Arrangements with Forbidden Positions Ex 8.7 : How many one-to-one funtions f: A B satisfy none of the following onditions: : f u or v : f w : f w or x : f x, y, or z B u v w x y z A rc, x x6x9x x 8xx 0x x P6, P5, 6

37 Arrangements with Forbidden Positions

38 Exerises You should explain your answers in detail, only one-line answer will be rated 0 8-: 6,, 6 8-: 5 8-: 6, 0 8-,5: 5, add f5 z How many integers n are suh that 0 n<0,000 and the sum of the digits is less than or equal to 7? 8

Math 378 Spring 2011 Assignment 4 Solutions

Math 378 Spring 2011 Assignment 4 Solutions Math 3 Spring 2011 Assignment 4 Solutions Brualdi 6.2. The properties are P 1 : is divisible by 4. P 2 : is divisible by 6. P 3 : is divisible by. P 4 : is divisible by 10. Preparing to use inclusion-exclusion,