1. Applications of Set Theory in Counting Techniques: Inclusion-Exclusion Rule

2. The Inclusion/Exclusion Rule for Two or Three Sets
If A, B and C are finite sets then
 n(A  B) = n(A) + n(B) – n(A  B)
 n(A  B  C) = n(A) + n(B) + n(C)
- n(A  B) – n(A  C) – n(B  C)
+ n(A  B  C) A B A B C

3. Example on Inclusion/Exclusion Rule (2 sets)
Question: How many integers from 1 through 100
are multiples of 3 or multiples of 7 ?
Solution: Let A=the set of integers from 1 through which are multiples of 3;
B = the set of integers from 1 through 100
which are multiples of 7.
Then we want to find n(A  B).
First note that A  B is the set of integers
from 1 through 100 which are multiples of 21 .
n(A  B) = n(A) + n(B) - n(A  B) (by incl./excl. rule)
= – 4 = 43 (by counting the elements
of the three lists)

4. Example on Inclusion/Exclusion Rule (3 sets)
3 headache drugs – A,B, and C – were tested on 40 subjects. The results of tests:
23 reported relief from drug A;
18 reported relief from drug B;
31 reported relief from drug C;
11 reported relief from both drugs A and B;
19 reported relief from both drugs A and C;
14 reported relief from both drugs B and C;
37 reported relief from at least one of the drugs.
Questions:
1) How many people got relief from none of the drugs?
2) How many people got relief from all 3 drugs?
3) How many people got relief from A only?

5. Example on Inclusion/Exclusion Rule (3 sets)
We are given: n(A)=23, n(B)=18, n(C)=31,
n(A  B)=11, n(A  C)=19, n(B  C)=14 ,
n(S)=40, n(A  B  C)=37
Q1) How many people got relief from none of the drugs?
By difference rule,
n((A  B  C)c ) = n(S) – n(A  B  C) = = 3 S A B C

6. Example on Inclusion/Exclusion Rule (3 sets)
Q2) How many people got relief from all 3 drugs?
By inclusion/exclusion rule:
n(A  B  C) = n(A  B  C)
- n(A) - n(B) - n(C)
+ n(A  B) + n(A  C) + n(B  C)
= 37 – 23 – 18 – = 9
Q3) How many people got relief from A only?
n(A – (B  C)) (by inclusion/exclusion rule)
= n(A) – n(A  B) - n(A  C) + n(A  B  C)
= 23 – 11 – = 2