I have a problem understanding this specific formula of inclusion and exclusion. (favorite) I need to know why it is needed. and an example that uses it.
\(\displaystyle \mathbf{Lemma 1. }\, Principle\, of\, inclusion\, and\, exclusion.\)
\(\displaystyle \mbox{Given a set }\, A,\, \mbox{ let }\, f\, \mbox{ be a real valued func}\mbox{tion}\)
\(\displaystyle \mbox{defined for all sets }\, S,\, J\, \subseteq\, A.\, \)
. . .\(\displaystyle \mbox{ If }\displaystyle g(S)\, =\, \sum_{J:J\, \supseteq\, S}\, f(J)\, \)
. . . . .\(\displaystyle \displaystyle \mbox{ then }\, f(S)\, =\, \sum_{J:J\, \supseteq\, S}(-1)^{\vert J\S \vert}\, g(J).\)
\(\displaystyle \mathbf{Lemma 1. }\, Principle\, of\, inclusion\, and\, exclusion.\)
\(\displaystyle \mbox{Given a set }\, A,\, \mbox{ let }\, f\, \mbox{ be a real valued func}\mbox{tion}\)
\(\displaystyle \mbox{defined for all sets }\, S,\, J\, \subseteq\, A.\, \)
. . .\(\displaystyle \mbox{ If }\displaystyle g(S)\, =\, \sum_{J:J\, \supseteq\, S}\, f(J)\, \)
. . . . .\(\displaystyle \displaystyle \mbox{ then }\, f(S)\, =\, \sum_{J:J\, \supseteq\, S}(-1)^{\vert J\S \vert}\, g(J).\)
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