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Michi’s blog » archive for January, 2009

 Homological Inclusion-Exclusion and the Mayer-Vietoris sequence

  • January 9th, 2009
  • 10:44 pm

This blogpost is inspired to a large part by comments made by Rob Ghrist, in connection to his talks on using the Euler characteristic integration theory to count targets detected by sensor networks.

He pointed out that the underlying principle inducing the rule
\chi(A\cup B) = \chi(A)+\chi(B)-\chi(A\cap B)
goes under many names, among those \emph{Inclusion-Exclusion}, favoured among computer scientists (and combinatoricists). He also pointed out that the origin of this principle is the Mayer-Vietoris long exact sequence
\cdots\to H_{n}(A\cap B)\to H_{n}(A)\oplus H_{n}(B)\to H_{n}(A\cup b)\to\cdots

In this blog post, I’d like to give more meat to this assertion as well as point out how the general principle of Inclusion-Exclusion for finite sets follows immediately from Mayer-Vietoris.

Inclusion-Exclusion, and the passage from two sets to many

The basic principle of Inclusion-Exclusion says that if we have two sets, A and B, then the following relationship of cardinalities holds:
|A\cup B| = |A| + |B| – |A\cap B