The Principle of Inclusion and Exclusion
Let’s begin with some comments about Figure 2.1, which is on the next page. We are looking at properties of subsets A, B, C of some universal set that has t elements. The universal set in the figure is represented by the rectangles. For each set S, we let |S| denote the number of elements in S and say that |S| is the size of S. Looking at the top third of the figure, we see a trivial result that is tremendously important. Frequently it is much easier to count the elements in a set by first counting the elements not in the set. For example, counting the 5-letter words that have a vowel is very difficult without first counting the number of 5-letter words that have no vowel. The number of 5-letter words that have a vowel is 265 – 215.
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