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 A_{1} and A_{2} be two sets. We want to show that
Note that the sets A_{1} and A_{2} might have some common elements. To be specific, the number of common elements between A_{1} and A_{2} is |A_{1} ∩ A_{2}|. Each of these elements is counted twice in |A_{1}| + |A_{2}| (once in |A_{1}| and once in |A_{2}|), although it should be counted as one element in |A_{1} ∪ A_{2}|. Therefore, the double count of these elements in |A_{1}| + |A_{2}| should be adjusted by the subtraction of the term |A_{1} ∩ A_{2}| 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 A_{1} denote the set of books that are novels, and A_{2} denote the set of books published in 1984. We have,
Consequently, according to eq. 1,
|A_{1} ∪ A_{2}| = 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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