Inclusion Exclusion of Finite Sets

We present in this section some results related to the cardinality of finite sets. We shall use the notation |P| to denote the cardinality of the set P. Some simple results, the derivation of which is left to the reader, are:



We show in the following a less obvious result. Let A₁ and A₂ be two sets. We want to show that



Note that the sets A₁ and A₂ might have some common elements. To be specific, the number of common elements between A₁ and A₂ is |A₁ ∩ A₂|. Each of these elements is counted twice in |A₁| + |A₂| (once in |A₁| and once in |A₂|), although it should be counted as one element in |A₁ ∪ A₂|. Therefore, the double count of these elements in |A₁| + |A₂| should be adjusted by the subtraction of the term |A₁ ∩ A₂| in the right hand side of (eqn. 1). As an example, suppose that among a set of 12 books, 6 are novels, 7 were published in the year 1984, and 3 are novels published in 1984. Let A₁ denote the set of books that are novels, and A₂ denote the set of books published in 1984. We have,



Consequently, according to eq. 1,

|A₁ ∪ A₂| = 6 + 7 – 3 = 10

That is, there are 10 books which are either novels or 1984 publications, or both. Consequently, among the 12 books there are 2 nonnovels that were not published in 1984.

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