Homework 6 Solutions - HOMEWORK 6 COMMENTS 1 Non-book problems(1 Negating statements with quantifiers(a We rewrite the statement as ∀> ∃

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Unformatted text preview: HOMEWORK 6: COMMENTS 1. Non-book 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 ....
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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.

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Homework 6 Solutions - HOMEWORK 6 COMMENTS 1 Non-book problems(1 Negating statements with quantifiers(a We rewrite the statement as ∀> ∃

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