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:
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAN8AAACJCAIAAAAjTHqRAAAGYklEQVR4nO2dW5rqIBAGXfNsx524Dlcxm/A8jEdJuKQhDfxJqt5GI4GmJMTM1317Aahym90BgCzYCbpgJ+iytPN5v91uP4/fSZ1p6cPzfrvlPtAwHN8ItLWmMAsalO38ffzc1tyf+8741inbnG1uwp5le9QwnBF2+kTgCtjsfAfw/5/ryP2FO4jy34FxgNcL3fvvxfyU5yac2c3vScNwLGbY7YmP3B+BK1FlZzqYZjtTF+G4wdLcRCcq0zCcvnbuj8C1qLMzmt7gY5t2vj+8fDVucdzamRpOVztdInAlWq7smY3Ulp3p+5do6Riw7ywMJ3/21W7xS6Gvq9ZcI3AFqu+KUlEbuHZmzm27ZzcMh7VTicp9Z5r3YZ+j1n+HrY/cdTUMh32nEi52LkOcu/6/otlJNr99V2S+vMrZuT8C18LJztfKnK29WOFI/bWzvQPBq2MicHCmPStK/zbl2wftZ0UjInBwJj7JDJeQ9d6sz5Xd2qlRTzIbInAt+C8Q0AU7QRfsBF2wE3TBTtAFO0EX7ARdsBN0wU7QBTtBF+wEXbATdMFO0AU7QRfsBF2wE3TBTtAFO0EX7ARdsBN0wU7QBTtBF+wEXbATdMFO0AU7QRfsBF1OXRGmbwOtRRFS5x+R7CmXkU84Y9OpK8LU9Kd2jtJZSsfUoHErNFNItSqBS0WY8L1CSrjhFWE6kklv2KkGTerkPoVmMmmqRXCpCGPMfTyyIsyn3a/O92f4HUpOZOJT6Z78vZ8Zon8Nmo2z7ApsbigKuFSEseWNH1oR5nNcaN273eXEJexc/pXoS3ZGGwK43859gVVePc9aESZp4LvV5TCSa2dxuIUJbQjgfjt3BjZXmE+As1aEcbQztrDezkIAZ6+dx7dzY9m32emw77QjZqdvHQXXfSd2Lg9sr4dizxvf1c72fWduTNFNfFGXzXv2qsDabzWHM9bO78FZrY6wdu64Z8/0NP7I5i3f1u+d1sAe9569I7nfpkb2YQfG3zutLa03keUmimepC6zyHfvJK8L0xPasyEDqBtxyrcqfxR7YYz0rggq6PKd+3ofJcrjn7ABKYCfogp2gC3aCLtgJumAn6IKdoAt2gi7YCbpgJ+iCnaALdoIu2Am6YCfogp2gC3aCLtgJumAn6IKdoAt2gi7YCbpgJ+iCnaALdoIu2Am6YCfoolcRpl9PHDJlzuhDj24Mbr+1J54VYaIcXFt5AQtHjrCzT7mWuj4Er3aNxmqsG8nJelQCCs5kHqlXRZj/Z10Xl8gnv/y2ksrpN9LO/eVa7L3dTAzbJxq/j5/P5xPZa9vnffWBpPvtI/WpCBNkKA3yT6ZzQ/rk5059u7e+4g2j627noGzlv4/HqrVi8kRzJaD4Xdd6QB4VYRYWLrKjJvLq9qltYKJhdL3tnBENwzCNlYAWb/aoB+RREWbh4DJ3b5yTPJ0VPfoSzbuyx1/ujSzDMfaU717RWHalGLf/h9Zf2YsX7C71gDwqwuQzEr8J7XRaLTyv7IXRnW3t/Ax52VjbvIcf7lMPyKOqQdWVfdROK0HD6M6y73y9Xlk1E+0bq1mE3e1SD8il5kbVXVHUz+q6MK0o2jksGp8LXLYyXmc7W0bqY2fdL0rh2zfjt9mDfna29yF4tWM00vug1fa3u53fg60j9XxWFO0/zVKPqQvT0OYgO8M3Z1TJ6d1++oTbI53+JDNUuvALnNuplO0cGo2KXnU75dZI+S8Q0AU7QRfsBF2wE3TBTtAFO0EX7ARdsBN0wU7QBTtBF+wEXbATdMFO0AU7QRfsBF2wE3TBTtAFO0EX7ARdsBN0wU7QBTtBF+wEXbATdMFO0AU7QRfsBF2m51FKYUo2ZO7mlPRJbS2IxF8Gz4owZlbJ6hoKoAQd25zMhkH1snP/wK+FV0UYM+sUuKlcefZJsljaMKgedvoO/Br4VIQx0ysRdSnvfsOg/O2cl4/8yHhUhLHjn8T/e61sXTt3FILZ6te86gUnwaMiTKbohqHK2+LVqkkKTmlKal83qEIHlmO1fiX8Bn4pPCrC2PFZQiq3Fw2DYu3UwKuqgZEZ268pxQzYd3ow2E6XAijpjUR2YZewc1jll1Mx3M7PaQpOnXDtDF4dNvDjM/NZ0aACKHrPiqZVfjkac59kthVA2Xllt/ar55PMSZVfjgb/BQK6YCfogp2gC3aCLv8Aw8H9ecndpXkAAAAASUVORK5CYII=)
We show in the following a less obvious result. Let A1 and A2 be two sets. We want to show that
![](data:image/png;base64,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)
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,
![](data:image/png;base64,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)
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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