Anonymous
Anonymous asked in Science & MathematicsAgriculture · 1 month ago

finite math question!!! please help?

Find a formula for the number of elements in the union of 4 sets that uses: 

n(A), n(B), n(C), n(D)

n(A⋂B), n(A⋂C), n(A⋂D), n(B⋂C), n(B⋂D), n(C⋂D)

n(A⋂B⋂C), n(A⋂C⋂D), n(B⋂C⋂D), n(A⋂B⋂D)

n(A⋂B⋂C⋂D)

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  • 1 month ago
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    By the inclusion-exclusion principle:

    n(A∪B∪C∪D) = 

    [n(A) + n(B) + n(C) + n(D)] 

    - [n(A⋂B) + n(A⋂C) + n(A⋂D) + n(B⋂C) + n(B⋂D) + n(C⋂D)] 

    + [n(A⋂B⋂C) + n(A⋂C⋂D) + n(B⋂C⋂D) + n(A⋂B⋂D)] 

    - n(A⋂B⋂C⋂D)

    You include the number of elements in A, B, C and D. 

    But you then have to exclude all combinations of two sets because those have been *overcounted*.

    In doing that, you need to include all combinations of three sets because those have now been *undercounted*.

    Finally, exclude the intersection of all 4 sets which again was *overcounted*.

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