By Steven R. Lay

Research with an creation to evidence, 5th variation is helping fill within the basis scholars have to achieve actual analysis-often thought of the main tough path within the undergraduate curriculum. via introducing common sense and emphasizing the constitution and nature of the arguments used, this article is helping scholars circulate rigorously from computationally orientated classes to summary arithmetic with its emphasis on proofs. transparent expositions and examples, valuable perform difficulties, quite a few drawings, and chosen hints/answers make this article readable, student-oriented, and instructor- pleasant. 1. common sense and evidence 2. units and services three. the true Numbers four. Sequences five. Limits and Continuity 6. Differentiation 7. Integration eight. endless sequence Steven R. Lay word list of key words Index

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B) x ∈ A or x ∈ B. (c) If x ∈ A, then x ∈ B. (d) If x ∉ A, then x ∈ B. 53 Sets and Functions 15. Which statement(s) below would enable one to conclude that x ∈ A ∩ B? (a) x ∈ A and x ∈ B. (b) x ∈ A or x ∈ B. (c) x ∈ A and x ∉ A\B. (d) If x ∈ A, then x ∈ B. 16. Which statement(s) below would enable one to conclude that x ∈ A \B? (a) x ∈ A and x ∉ B \A. (b) x ∈ A ∪ B and x ∉ B. (c) x ∈ A ∪ B and x ∉ A ∩ B. (d) x ∈ A and x ∉ A ∩ B. 17. Which statement(s) below would enable one to conclude that x ∉ A \B?

1. Mark each statement True or False. Justify each answer. (a) When an implication p ⇒ q is used as a theorem, we refer to p as the antecedent. (b) The contrapositive of p ⇒ q is ~ p ⇒ ~ q. 26 Logic and Proof (c) The inverse of p ⇒ q is ~ q ⇒ ~ p. (d) To prove “∀ n, p (n) ” is true, it takes only one example. (e) To prove “∃ n p (n) ” is true, it takes only one example. 2. Mark each statement True or False. Justify each answer. (a) When an implication p ⇒ q is used as a theorem, we refer to q as the conclusion.

What is the common name for a member of Ex? Suppose that y is an atom with six protons. What is the common name for a member of Ey?

### Analysis with an introduction to proof by Steven R. Lay

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