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  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
( ) – ( ) + ( ) – (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 – – 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
|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?