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 ) where n( 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

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 million – n( P ∩ S ) 245 million = 264 million – n( P ∩ S ) – 19 million = – n( P ∩ S ) n( P ∩ S ) = 19 million

Roman Numerals Arabic NumeralsRoman Numerals 1I 2II 3III 4IV 5V 6VI 7VII 8VIII

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) S I II U

Shade S S I II U

Shade S ´ S I II 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´) S I II U III T IV

Shade T S I II U III T IV

Shade T ´ S I II U III T IV

Three Set Venn Diagram Sets R, S and T S I II U III T IV R V 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´