Chapter 5 Section 2 Fundamental Principle of Counting

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Chapter 5 Section 2 Fundamental Principle of Counting

Chapter 5 Section 2 Fundamental Principle of Counting

Definition & Notation • Definition: – Combinatorics : The mathematical field dealing with 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

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 ∩

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 ) =

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}

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

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 ) =

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(

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 ) = 19 million

Roman Numerals Arabic Numerals 1 2 3 4 5 6 7 8 Roman Numerals

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

Single Set Venn Diagram • Single Set S II Two basic regions: Basic region

Single Set Venn Diagram • Single Set S II Two basic regions: Basic region I = S Basic region II = S´ U I S (in set S) (not in set S)

Shade S U II I S

Shade S U II I S

Shade S´ U II S I

Shade S´ U II S I

Two Set Venn Diagram • Sets S and T IV U S II I

Two Set Venn Diagram • Sets S and T IV U S II I T III Four basic regions are: Basic region I: (S T), Basic Region II: (S T´) Basic region III: (S´ T), Basic Region IV: (S´ T´)

Shade T U IV S II I T III

Shade T U IV S II I T III

Shade T ´ U IV S II I T III

Shade T ´ U IV S II I T III

Three Set Venn Diagram U • Sets R , S and T R II

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

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

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´

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