The Sieve Methods
The Sieve Methods
PIE (Principle of Inclusion-Exclusion) AUB|=A+|B-A∩B AUBUC=A+B+C -A∩B-A∩C1-|BnC1 +|A∩B∩C C BnC AnC AnBnC A B AnB
PIE (Principle of Inclusion-Exclusion) |A B| = |A| + |B||A ⇥ B| |A B C| = |A| + |B| + |C| |A ⇥ B| |A ⇥ C| |B ⇥ C| +|A B C|
PIE (Principle of Inclusion-Exclusion) a-A-,之4n+ 三1 1<i<n 1≤i<j≤n …+(-1)n-1|A1∩.∩An ≥-11门4 IC{1,,n} I≠0
PIE (Principle of Inclusion-Exclusion) ⇥ n i=1 Ai = 1in |Ai| 1i<jn |Ai ⇥ Aj |+ ··· + (1)n1|A1 ⇤ ··· ⇤ An| = I{1,...,n} I= (1)|I|1 iI Ai
PIE (Principle of Inclusion-Exclusion) A1,A2,.,AnCU<— universe 西nena=-4刘 2=1 --- ic T I≠0 Ar=∩Ai Ao=U i∈I
PIE (Principle of Inclusion-Exclusion) A1, A2,...,An U universe A1 ⇥ A2 ⇥ ··· An = U ⇥ n i=1 Ai AI = iI Ai A = U = |U| I{1,...,n} I= (1)|I|1 iI Ai
PIE (Principle of Inclusion-Exclusion) A1,A2,...,An CUuniverse AnAn…An=(-1)川|Ar IC{1,,n} AI=∩Aa Ao=U i∈I
PIE (Principle of Inclusion-Exclusion) A1, A2,...,An U universe A1 ⇥ A2 ⇥ ··· An = AI = iI Ai A = U I{1,...,n} (1)|I| |AI |
PIE (Principle of Inclusion-Exclusion) A1,A2,...,An U universe AnAn…An=S0-S1+S2+…+(-1)nSm A1=∩A, Ao=U i∈I Sk=∑|A So=Ao=U I=k
PIE (Principle of Inclusion-Exclusion) A1, A2,...,An U universe A1 ⇥ A2 ⇥ ··· An = AI = iI Ai A = U Sk = |I|=k |AI | S0 = |A| = |U| S0 S1 + S2 + ··· + (1)nSn
Surjections of f (nl onto,(ml U=[m→[m A=[m→([m\{}) ∩A=∑(-1)1A iElm] IC[m] AI=∩A: Ao=U i∈I
Surjections f : [n] onto ⇥ [m] # of U = [n] [m] Ai = [n] ([m] \ {i}) AI = iI Ai A = U I[m] (1)|I| |AI | i[m] Ai =
Surjections U=[m]→[m A:=[m→(m\{i) Ao=U Ar=∩Ai=[m→(Im\I) i∈I |Ar=(m-1I)” ∩A, =>(-1)|A iElm] IC[m] ∑(-1(m-1W”=-1()m- = IC[m] =-m-(）
Surjections U = [n] [m] Ai = [n] ([m] \ {i}) AI = iI A = U Ai = [n] ([m] \ I) |AI | = (m |I|) n = ⇤ m k=1 (1)mk m k ⇥ kn = I[m] (1)|I| (m |I|) n I[m] (1)|I| |AI | i[m] Ai = = m k=0 (1)k m k (m k) n
Surjections 网n四=-*（四） m kn k=1 (f-1(0),f-1(1),.,f-1(m-1) ordered m-partition of [n] 例-m{h} {}=-(） kn
Surjections = ⇤ m k=1 (1)mk m k ⇥ kn [n] onto ⇥ [m] (f 1(0), f 1(1),...,f 1(m 1)) ordered m-partition of [n] [n] onto [m] = m! n m n m = 1 m! m k=1 (1)mk m k kn
Derangement les problemes des rencontres ESSAY DANALYSE Two decks,A and B,of cards: 5U我 LES JEUX DE HAZARD. The cards of A are laid out in a row, Par M:Remond de Montmort. and those of B are placed at random, one at the top on each card of A. w光a What is the probability that A PARIS, Chez JAcov Qo,Imprimeur-Jure-Libraire de I'Univerlite,rue Galande. no 2 cards are the same in each pair? M D CC V I I L AVEC APPROBATION ET PRINILIGE DU ROY
Derangement Two decks, A and B, of cards: The cards of A are laid out in a row, and those of B are placed at random, one at the top on each card of A. What is the probability that no 2 cards are the same in each pair? les problèmes des rencontrés :