Chapter 5 Section 2

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( SUT ) = n( S ) + n( T ) – n( S∩T ) where n( SUT ) 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( SUT ) = 13 • Find n( S )

Exercise 5 Solution • Inclusion – Exclusion Formula n( SUT ) = 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( PUS ) = 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 U • Single Set S Two basic regions: Basic region I = S (in set S) Basic region II = S´ (not in set S) S II I

Shade S U S II I

Shade S´ U S II I

Two Set Venn Diagram U • 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´) IV S T I III II

Shade T U IV S T I III II

Shade T ´ U IV S T I III II

Three Set Venn Diagram U • Sets R , S and T R S II VI V I III IV T VIII VII

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´

Chapter 5 Section 2