Applications of set theory in counting techniques inclusion exclusion rule
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Applications of Set Theory in Counting Techniques: Inclusion-Exclusion Rule. C. A. A. B. B. 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)

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The inclusion exclusion rule for two or three sets

C

A

A

B

B

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)


Example on inclusion exclusion rule 2 sets
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 100 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)

    = 33 + 14 – 4 = 43 (by counting the elements

    of the three lists)


Example on inclusion exclusion rule 3 sets
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?


Example on inclusion exclusion rule 3 sets1

C

A

B

Example on Inclusion/Exclusion Rule (3 sets)

S

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) = 40 - 37 = 3


Example on inclusion exclusion rule 3 sets2
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 – 31 + 11 + 19 + 14 = 9

Q3)How many people got relief from A only?

n(A – (B  C)) (byinclusion/exclusion rule)

= n(A) – n(A  B) - n(A  C) + n(A  B  C)

= 23 – 11 – 19 + 9 = 2