Unit 3-Lecture 5: The Pigeonhole Principle and Inclusion-Exclusion

2024-05-16 18:34:12

1 The Pigeonhole Principle

Pigeonhole Principle:
If there are more pigeons than holes they occupy, then at least two pigeons must be in the same hole.
Generalized Pigeonhole Principle:
If |a|>k⋅|b| |a| > k \cdot |b| , then every total function f：A→B f：A \to B maps at least k+1 k + 1 different elements of A to the same element of B.

2 Inclusion-Exclusion

Union of n Sets:

|S1∪S2∪...∪Sn|=∑ϕ≠I⊆1...n(−1)|I|+1|⋂i∈ISi|

|S_1 \cup S_2 \cup...\cup S_n| = \sum_{\phi \neq I \subseteq {1...n}}(-1)^{|I|+1}|\bigcap_{i\in I}S_i|

Reference

[1] Lehman E, Leighton F H, Meyer A R. Mathematics for Computer Science[J]. 2015.

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