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 |