1.2.2: Inclusion Exclusion Principle

Finite Math (SJDC M20) v2 · LibreTexts

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    From LibreTexts, under not declared, as the source states it; its equations are written in TeX, which the PDF shows typeset. This page at LibreTexts.

    The Inclusion-Exclusion Principle is a counting method used when groups overlap. It helps you count the total number of distinct items without counting any item more than once.

    Two Sets

    If you have two sets, A and B:

    • Start by adding the number of elements in each set.
    • Because any element that belongs to both sets is counted twice, subtract the overlap once.

    n(A ∪ B) = n(A) + n(B) − n(A ∩ B)

    ChatGPT Image Jul 30, 2026, 04_27_52 PM.png

     

    Example:

    In a class:

    • 18 students play soccer.
    • 12 students play basketball.
    • 5 students play both.

    The total number of students who play at least one of the sports is

    18+12−5=25.18 + 12 - 5 = 25.18+12−5=25.

    The 5 students who play both sports were counted twice, so we subtract them once.

    Three Sets

    With three sets, the idea is similar, but there is one extra step.

    • Add the sizes of all three sets.
    • Subtract each pairwise overlap.
    • Add back the overlap of all three sets because it was subtracted too many times.

    ChatGPT Image Jul 30, 2026, 04_25_54 PM.png

    1.2.2: Inclusion Exclusion Principle is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.