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Counting Elements of Disjoint Sets: The Addition Rule

Counting Elements of Disjoint Sets: The Addition Rule. Lecture 30 Sections 6.3 Tue, Mar 20, 2007. Example: Inclusion/Exclusion. How many primes are there between 1 and 100? The non-primes must be multiples of 2, 3, 5, or 7, since the square root of 100 is 10. A Lemma.

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Counting Elements of Disjoint Sets: The Addition Rule

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  1. Counting Elements of Disjoint Sets: The Addition Rule Lecture 30 Sections 6.3 Tue, Mar 20, 2007

  2. Example: Inclusion/Exclusion • How many primes are there between 1 and 100? • The non-primes must be multiples of 2, 3, 5, or 7, since the square root of 100 is 10.

  3. A Lemma • Lemma: Let n and d be positive integers. There are n/d multiples of d between 1 and n, where x represents the “floor” of x.

  4. Example: Inclusion/Exclusion • Let A = {n | 1 n 100 and 2 divides n}. • Let B = {n | 1 n 100 and 3 divides n}. • Let C = {n | 1 n 100 and 5 divides n}. • Let D = {n | 1 n 100 and 7 divides n}.

  5. Example: Inclusion/Exclusion • By the Inclusion/Exclusion Rule, |ABCD| = |A| + |B| + |C| + |D| – |A B| – |A C| – |A D| – |B C| – |B D| – |C D| + |A B C| + |A B C| + |A B C| + |A B C| – |A B C  D|.

  6. Example: Inclusion/Exclusion • However, • A B = {n | 1 n 100 and 6 | n}. • A B C = {n | 1 n 100 and 30 | n}. • B C D = {n | 1 n 100 and 105 | n}. • And so on.

  7. Example: Inclusion/Exclusion • Therefore, • |A| = 100/2 = 50. • |A B| = 100/6 = 16. • |A B C| = 100/30 = 3. • |B C D| = 100/105 = 0. • And so on.

  8. Example: Inclusion/Exclusion • The number of multiples of 2, 3, 5, and 7 is (50 + 33 + 20 + 14) – (16 + 10 + 7 + 6 + 4 + 2) + (3 + 2 + 1 + 0) – (0) = 78

  9. Example: Inclusion/Exclusion • This count includes 2, 3, 5, 7, which are prime. • This count does not include 1, which is not prime. • Therefore, the number of primes is 100 – 78 + 4 – 1 = 25.

  10. Primes.cpp • Primes.cpp

  11. Example: Inclusion/Exclusion • How many integers from 1 to 1000 are multiples of 6, 10, or 15? • Let A = {n | 1 n 100 and 6 divides n}. • Let B = {n | 1 n 100 and 10 divides n}. • Let C = {n | 1 n 100 and 15 divides n}. • What is A B? A C? B C? • What is A B C?

  12. Example: Inclusion/Exclusion • |A| = 1000/6 = 166. • |B| = 1000/10 = 100. • |C| = 1000/15 = 66. • |A B| = 1000/30 = 33. • |A C| = 1000/30 = 33. • |B C| = 1000/30 = 33. • |A B C| = 1000/30 = 33. • Therefore, 266 numbers from 1 to 1000 are multiples of 6, 10, or 15.

  13. Example: Inclusion/Exclusion • How many 8-bit numbers have either • 1 in the 1st and 2nd positions, or • 1 in the 1st and 3rd positions, or • 1 in the 2nd, 3rd, and 4th positions?

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