This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: HOMEWORK 6: COMMENTS 1. Nonbook problems (1) Negating statements with quantifiers. (a) We rewrite the statement as ∀ > , ( ∃ δ > , ( ∀ x ∈ R , (  x a  < δ ⇒  f ( x ) f ( a )  < ))) which is the same as ∀ ∈ R > , ( ∃ δ ∈ R > , ( ∀ x ∈ R , (  x a  < δ ⇒  f ( x ) f ( a )  < ))) . The negation is ∃ ∈ R > , ( ∀ δ ∈ R > , ( ∃ x ∈ R , (  x a  < δ and  f ( x ) f ( a )  ≥ ))) In english: there is an > 0 such that for every δ > 0, we have  x a  < δ for some x ∈ R and  f ( x ) f ( a )  ≥ . (b) We rewrite the statement as ∀ M > , ( ∃ δ > , ( ∀ x ∈ R , ( < x a < δ ⇒ f ( x ) > M ))) which is the same as ∀ M ∈ R > , ( ∃ δ ∈ R > , ( ∀ x ∈ R , ( < x a < δ ⇒ f ( x ) > M ))) . The negation is ∃ M ∈ R > , ( ∀ δ ∈ R > , ( ∃ x ∈ R , ( < x a < δ and f ( x ) ≤ M ))) In english: there is an M > 0 such that for every δ > 0, there is some x ∈ R for which 0 < x a < δ and f ( x ) ≤ M ....
View
Full
Document
This note was uploaded on 06/15/2009 for the course MATH 74 taught by Professor Courtney during the Spring '07 term at University of California, Berkeley.
 Spring '07
 COURTNEY
 Division

Click to edit the document details