To use the inclusion/exclusion principle to obtain |A U B|, we need |A|, |B| and |A ∩ B|. A ∩ B = The set of all integers that are both multiples of 3 and 5, which also is the set of integers that are multiples of 15.A U B = The set of all integers from 1 to 1000 that are multiples of either 3 or 5.Let us assume that B = set of all integers from 1 to 1000 that are multiples of 5.Let us assume that A = set of all integers from 1 to 1000 that are multiples of 3.Example 1: How many integers from 1 to 1000 are either multiples of 3 or multiples of 5?.Hence, we have |B| = 14.) and the multiple of 21… n(A ∩ B) = 4Įxample: Inclusion and Exclusion Principle Similarly for multiples of 7, each multiple of 7 is of the form 7q for some integer q from 1 through 14.From 1 to 100, every third integer is a multiple of 3, each of this multiple can be represented as 3p, for any integer p from 1 through 33, Hence |A| = 33.= 33 + 14 – 4 = 43 (by counting the elements.First note that A ∩ B is the set of integers from 1 through 100 which are multiples of 21. Solution: Let A=the set of integers from 1 through 100 which are multiples of 3 B = the set of integers from 1 through 100 which are multiples of 7.Question: How many integers from 1 through 100 are multiples of 3 or multiples of 7 ?.C(8,3) = 56…thus by inclusion and exclusion principle Įxample on Inclusion/Exclusion Rule (2 sets).Let A be the set of committees that include Professor-A and B be the set of committees that include Professor-B.Total number of committees possible is C(10,5) = 252 ways Then by the PIE we have | F∩R | = |F| + |R| - |F U R|.n (F∩R) = | F∩R | = 20 and we need to find |F U R|.If F is faculty who speak French and R is the faculty who speak Russian, then given that Inclusion-Exclusion U A A B It’s simply a matter of not over-counting the blue area in the intersection. Section 3.3 Principle of Inclusion & Exclusion Pigeonhole Principle 4 This principle can be generalized to n sets.The name comes from the fact that to calculate the elements in a union, we include the individual elements of A and B but subtract the elements common to A and B so that we don’t count them twice.Principle of Inclusion & Exclusion The above equation represents the principle of inclusion and exclusion for two sets A and B. The Principle of Inclusion and Exclusion (PIE) is a counting technique that computes the number of elements that satisfy at least one of several properties while guaranteeing that elements satisfying more than one property are not counted twice.
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