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5: The Principle of Inclusion and Exclusion

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    6114
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    One of our very first counting principles was the sum principle which says that the size of a union of disjoint sets is the sum of their sizes. Computing the size of overlapping sets requires, quite naturally, information about how they overlap. Taking such information into account will allow us to develop a powerful extension of the sum principle known as the “principle of inclusion and exclusion.”

    • 5.1: The Size of a Union of Sets
      One of our very first counting principles was the sum principle which says that the size of a union of disjoint sets is the sum of their sizes. Computing the size of overlapping sets requires, quite naturally, information about how they overlap. Taking such information into account will allow us to develop a powerful extension of the sum principle known as the “principle of inclusion and exclusion.”
    • 5.2: Applications of Inclusion and Exclusion
      We defined a graph to consist of set V of elements called vertices and a set E of elements called edges such that each edge joins two vertices. A coloring of a graph by the elements of a set C (of colors) is an assignment of an element of C to each vertex of the graph; that is, a function from the vertex set V of the graph to C. A coloring is called proper if for each edge joining two distinct vertices, the two vertices it joins have different colors.
    • 5.3: Deletion-Contraction and the Chromatic Polynomial
      In Chapter 2 we introduced the deletion-contraction recurrence for counting spanning trees of a graph. In this section, we will use the deletion-contraction recurrence to reduce the computation of the chromatic polynomial of a graph (exemplified by Figure 5.3.1) to the computation of chromatic polynomials that can easily be computed.
    • 5.4: The Principle of Inclusion and Exclusion (Exercises)
      This section contains the supplementary problems related to the materials discussed in Chapter 5.

    Thumbnail: Inclusion–exclusion illustrated by a Venn diagram for three sets. (CC BY-SA 3.0; Wikipedia).

    Contributors and Attributions


    This page titled 5: The Principle of Inclusion and Exclusion is shared under a GNU Free Documentation License 1.3 license and was authored, remixed, and/or curated by Kenneth P. Bogart.

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