Counting Elements of Disjoint Sets: The Addition Rule

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Presentation transcript:

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 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.

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}.

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  D|.

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.

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.

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

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.

Primes.cpp Primes.cpp

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?

Example: Inclusion/Exclusion |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.

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?