1. Counting Elements of Disjoint Sets: The Addition Rule
Lecture 26
Section 6.3
Fri, Feb 25, 2005

2. 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|.

3. 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|.

4. Counting Elements in Subsets
Corollary: Let A and B be finite sets with B  A. Then
|B| = |A| – |A – B|.
Corollary: Let S be the sample space of an experiment and let E be an event. Then
P(Ec) = 1 – P(E).

5. Counting Elements in Unions of Sets
Theorem: Let A and B be any finite sets. Then
|A  B| = |A| + |B| – |A  B|.
Proof:
(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|.

6. P(A  B) = P(A) + P(B) – P(A  B).
Probability
Corollary: Given an experiment, let A and B be two events. Then
P(A  B) = P(A) + P(B) – P(A  B).
Example: Draw two cards from a deck. What is the probability that at least one of them is black?

7. 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|.

8. 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 = , so there are 61  31 = 1891 divisors of 2030.

9. 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.
Thus, – 1271 = 2301 numbers divide either 1040 or 2030.

10. 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|.
The pattern is to add “one at a time”, subtract “two at a time”, and add “three at a time.”

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

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

13. 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)|.

14. 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)|.

15. 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|.

16. 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|

17. 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|

18. Number of Elements in the Union of Three Sets
Corollary: Let A, B, and C be any three events. Then
P(A  B  C) = P(A) + P(B) + P(C)
– P(A  B) – P(A  C) – P(B  C)
+ P(A  B  C).

19. 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|.