Chapter 5 Section 2

Fundamental Principle of Counting

Definition & Notation

• Definition:– Combinatorics : The mathematical field dealing

with counting problems

• Notation:– Notation to represent the number of

elements in a set S : n ( S )

Inclusion – Exclusion Principle

• Formula: n( S U T ) = n( S ) + n( T ) – n( S ∩ T )

wheren( S U T ) is the number of element in the union of

sets S and T.n( S ) is the number of elements in set S.n( T ) is the number of elements in set T.n( S ∩ T ) is the number of element in the both

sets S and T.

Exercise 5 (page 217)

• Given:

n( T ) = 7

n( S ∩ T ) = 5

n( S U T ) = 13

• Find n( S )

Exercise 5 Solution

• Inclusion – Exclusion Formula

n( S U T ) = n( S ) + n( T ) – n( S ∩ T )

Using substitution

( 13 ) = n( S ) + ( 7 ) – ( 5 )

13 = n( S ) + 2

n( S ) = 11

n( S ) = 11

Exercise 9 (page 217)

• Let– U = { Adults in South America}– P = { Adults in South America who are fluent

in Portuguese }– S = { Adults in South America who are fluent

in Spanish }

Exercise 9 (page 217)

• Given:– 245 million are fluent in Portuguese or Spanish

(or both)– 134 million are fluent in Portuguese– 130 million are fluent in Spanish

• Find the number who are fluent in both (Portuguese and Spanish)

Exercise 9 Given

Using mathematical Notation

• n( P U S ) = 245 million

• n( P ) = 134 million

• n( S ) = 130 million

• Find n( P ∩ S )

Exercise 9 Solution

n( P ∩ S ) = n( P ) + n( S ) – n( P ∩ S )

245 million = 134 million + 130 million – n( P ∩ S )

245 million = 264 million – n( P ∩ S )

– 19 million = – n( P ∩ S )

n( P ∩ S ) = 19 million

Roman Numerals

Arabic Numerals Roman Numerals

1 I

2 II

3 III

4 IV

5 V

6 VI

7 VII

8 VIII

Single Set Venn Diagram

• Single Set S

Two basic regions:

Basic region I = S (in set S)

Basic region II = S´ (not in set S)

SIII

U

Shade S

SIII

U

Shade S´

SIII

U

Two Set Venn Diagram

• Sets S and T

Four basic regions are:

Basic region I: (S T), Basic Region II: (S T´)

Basic region III: (S´ T), Basic Region IV: (S´ T´)

SI

II

U

IIIT

IV

Shade T

SI

II

U

III

TIV

Shade T ´

S I

II

U

IIIT

IV

Three Set Venn Diagram

• Sets R , S and T

S

I

II

U

III

T

IV

RV VI

VIIVIII

Set Notation for the Basic Regions in a Three Set Venn diagram

• Basic region I: R S T

• Basic region II: R S T´

• Basic region III: R´ S T

• Basic region IV: R S´ T

• Basic region V: R S´ T´

• Basic region VI: R´ S T´

• Basic region VII: R´ S´ T

• Basic region VIII: R´ S´ T´