Past exam of the year 1 of the computer science course of the University of Oxford / 2022 / Discrete mathematics, continuous mathematics, and probability / Discrete mathematics / 1 / c / i

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- Solution i

Solution

For finite sets, the inclusion-exclusion principle states

i = 1 ⋃ n A_{i} = \emptyset \neq = J \subseteq {1, \dots, n} \sum (- 1)^{∣ J ∣ + 1} j \in J ⋂ A_{j} .

Thus single-set cardinalities are added, pairwise-intersection cardinalities subtracted, triple-intersection cardinalities added, and so on.

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