MA375: The Principle of Induction

Lecture Notes

Inclusion-Exclusion Principle (Basic)

Let B and C be subsets of a given set A. To count the number of elements in the union of B and C, we must evaluate the following:

$ |B \cup C| = |B| + |C| - |B \cap C| $

Subtracting $ |B \cap C| $ corrects the overcount.

In general,

$ \displaystyle |A_1 \cup A_2 \cup ... \cup A_n| = $

$ \displaystyle |A_1| + |A_2| + ... + |A_n| $

$ \displaystyle - |A_1 \cap A_2| - |A_1 \cap A_3| - ... - |A_(n-1)\cap A_n| $

$ \displaystyle + |A_1 \cap A_2 \cap A_3| + |A_1 \cap A_2 \cap A_4| + ... + |A_(n-2) \cap A_(n-1) \cap A_n| $

$ \displaystyle + (-1)^(n+1)|A_1 \cap A_2 \cap A_3 \cap ... \cap A_n| $

Back to MS375, Fall 2008, Prof. Walther