Counting Elements of Disjoint Sets: The Addition Rule Lecture 29 Section 6.3 Mon, Mar 19, 2007

Counting Elements in Disjoint Sets Theorem: Let {A1, …, An} be a partition of a set A. Then |A| = |A1| + … + |An|. Corollary: Let {A1, …, An} be a collection of pairwise disjoint finite sets. Then |A1  …  An| = |A1| + … + |An|.

Counting Elements in Subsets Theorem: Let A and B be finite sets with B  A. Then |A – B| = |A| – |B|. Proof: {B, A – B} is a partition of A. Therefore, |B| + |A – B| = |A|. So, |A – B| = |A| – |B|.

Counting Elements in Unions of Sets Theorem: Let A and B be any finite sets. Then |A  B| = |A| + |B| – |A  B|. Proof: One can verify that (A  B) – B = A – (A  B). Furthermore, B  A  B and A  B  A. Therefore, |A  B| – |B| = |A| – |A  B|. So, |A  B| = |A| + |B| – |A  B|.

Putnam Problem A-1 (1983) How many positive integers n are there such that n is an exact divisor of at least one of the numbers 1040, 2030? Let A = {n | n divides 1040}. Let B = {n | n divides 2030}. Then |A  B| = |A| + |B| – |A  B|.

Putnam Problem A-1 (1983) Prime factorization: 1040 = 240  540. Therefore, n | 1040 if and only if n = 2a5b where 0  a  40 and 0  b  40. There are 41  41 = 1681 such numbers. Similarly, 2030 = 260530, so there are 61  31 = 1891 divisors of 2030.

Putnam Problem A-1 (1983) Finally, an integer is in A  B if it divides both 1040 and 2030. That means that it divides the gcd of 1040 and 2030. The gcd of 240  540 and 260  530 is 240  530. Therefore, there are 41  31 = 1271 such numbers.

Putnam Problem A-1 (1983) Thus, 1681 + 1891 – 1271 = 2301 numbers divide either 1040 or 2030.

Number of Elements in the Union of Three Sets Theorem: Let A, B, and C be any three finite sets. Then |A  B  C| = |A| + |B| + |C| – |A  B| – |A  C| – |B  C| + |A  B  C|. Add sets one at a time, Subtract sets two at a time, Add sets three at a time.

Number of Elements in the Union of Three Sets Theorem: Let A, B, and C be any three finite sets. Then |A  B  C| = |A| + |B| + |C| – |A  B| – |A  C| – |B  C| + |A  B  C|. Add sets one at a time, Subtract sets two at a time, Add sets three at a time.

Number of Elements in the Union of Three Sets Theorem: Let A, B, and C be any three finite sets. Then |A  B  C| = |A| + |B| + |C| – |A  B| – |A  C| – |B  C| + |A  B  C|. Add sets one at a time, Subtract sets two at a time, Add sets three at a time.

Proof, continued Proof: |A  B  C| = |A  B| + |C| – |(A  B)  C|.

Proof, continued Proof: |A  B  C| = |A  B| + |C| – |(A  B)  C| = |A| + |B| – |A  B| + |C| – |(A  B)  C|.

Proof, continued Proof: |A  B  C| = |A  B| + |C| – |(A  B)  C| = |A| + |B| – |A  B| + |C| – |(A  B)  C| = |A| + |B| – |A  B| + |C| – |(A  C)  (B  C)|.

Proof, continued Proof: |A  B  C| = |A  B| + |C| – |(A  B)  C| = |A| + |B| – |A  B| + |C| – |(A  B)  C| = |A| + |B| – |A  B| + |C| – |(A  C)  (B  C)| = |A| + |B| – |A  B| + |C| – |A  C| – |B  C| + |(A  C)  (B  C)|.

Proof, continued Proof: |A  B  C| = |A  B| + |C| – |(A  B)  C| = |A| + |B| – |A  B| + |C| – |(A  B)  C| = |A| + |B| – |A  B| + |C| – |(A  C)  (B  C)| = |A| + |B| – |A  B| + |C| – |A  C| – |B  C| + |(A  C)  (B  C)| + |A  B  C|.

Proof, continued Proof: |A  B  C| = |A  B| + |C| – |(A  B)  C| = |A| + |B| – |A  B| + |C| – |(A  B)  C| = |A| + |B| – |A  B| + |C| – |(A  C)  (B  C)| = |A| + |B| – |A  B| + |C| – |A  C| – |B  C| + |(A  C)  (B  C)| + |A  B  C|. = |A| + |B| + |C| – |A  B| – |A  C| – |B  C|

Proof, continued Proof: |A  B  C| = |A  B| + |C| – |(A  B)  C| = |A| + |B| – |A  B| + |C| – |(A  B)  C| = |A| + |B| – |A  B| + |C| – |(A  C)  (B  C)| = |A| + |B| – |A  B| + |C| – |A  C| – |B  C| + |(A  C)  (B  C)| + |A  B  C|. = |A| + |B| + |C| – |A  B| – |A  C| – |B  C|

The Inclusion/Exclusion Rule Theorem: Let A1, …, An be finite sets. Then |A1  …  An| = i |Ai| – i j > i |Ai  Aj| + i j > i k > j |Ai  Aj  Ak| :  |A1  …  An|.

The Inclusion/Exclusion Rule The Inclusion/Exclusion Rule can be proved by induction.

Number of Elements in the Union of Four Sets Let U be the set of all pairs of distinct cards from a deck of 52 playing cards. How many pairs are there in which at least one of the two cards is black or a face card?

Number of Elements in the Union of Four Sets Let A = all pairs where 1st card is black. B = all pairs where 1st card is a face card. C = all pairs where 2nd card is black. D = all pairs where 2nd card is a face card. Find the number of elements in |A  B  C  D|.

Number of Elements in the Union of Four Sets How many pairs are there in which at least one of the two cards is black or a face card?

The Inclusion/Exclusion Rule Suppose five sets intersect as indicated in the following Venn diagram. A B C D E

The Inclusion/Exclusion Rule State the equation of the inclusion/exclusion rule for these sets. A B C D E

The Inclusion/Exclusion Rule Do the same for these sets. A B C D