By Steven R. Lay
Research with an creation to facts, 5th variation is helping fill within the foundation scholars have to reach genuine analysis-often thought of the main tricky path within the undergraduate curriculum. by means of introducing good judgment and emphasizing the constitution and nature of the arguments used, this article is helping scholars circulation rigorously from computationally orientated classes to summary arithmetic with its emphasis on proofs. transparent expositions and examples, invaluable 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 features three. the genuine Numbers four. Sequences five. Limits and Continuity 6. Differentiation 7. Integration eight. limitless sequence Steven R. Lay thesaurus of key words Index
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Which of the following statements are true? 2(d) we found that the collection D of all prime numbers between 8 and 10 is a legitimate set. This is so because the statement “ x ∈ D ” is always false, since there are no prime numbers between 8 and 10. Thus D is an example of the empty set, a set with no members. It is not difficult to show (Exercise 18) that there is only one empty set, and we denote it by ∅. For our first theorem we shall prove that the empty set is a subset of every set. Notice the essential role that definitions play in the proof.
35 Logic and Proof 14. Prove: If x/(x – 2) ≤ 3, then x < 2 or x ≥ 3. 15. Prove: log 2 7 is irrational. 16. Prove: If x is a real number, then | x + 1 | ≤ 3 implies that − 4 ≤ x ≤ 2. 17. ” (a) Suppose m is odd. Then m = 2k + 1 for some integer k. Thus m2 = (2k + 1)2 = 4k2 + 4k + 1 = 2(2k2 + 2k) +1, which is odd. Thus if m2 is odd, then m is odd. (b) Suppose m is not odd. Then m is even and m = 2k for some integer k. Thus m2 = (2k)2 = 4k2 = 2(2k2), which is even. Thus if m is not odd, then m2 is not odd.
13. Suppose you are to prove that sets A and B are disjoint. Write a reasonable beginning sentence for the proof, and indicate what you would have to show in order to finish the proof. 14. 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) 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.
Analysis with an introduction to proof by Steven R. Lay