By Steven R. Lay

Research with an creation to evidence, 5th variation is helping fill within the basis scholars have to achieve genuine analysis-often thought of the main tough direction within the undergraduate curriculum. by means of 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, precious perform difficulties, quite a few drawings, and chosen hints/answers make this article readable, student-oriented, and instructor- pleasant. 1. good judgment and facts 2. units and services three. the genuine Numbers four. Sequences five. Limits and Continuity 6. Differentiation 7. Integration eight. limitless sequence Steven R. Lay word list of keywords Index

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**Example text**

27 Logic and Proof (f) (g) (h) (i) If p is even or q is even, then pq is even. If pq is odd, then p is odd and q is odd. If p2 is even, then p is even. If p2 is odd, then p is odd. 8. Let f be the function given by f (x) = 4x + 7. Use the contrapositive implication to prove the statement: If x1 ≠ x2, then f (x1) ≠ f (x2). 9. In each part, a list of hypotheses is given. These hypotheses are assumed to be true. 12, you are to establish the desired conclusion. Indicate which tautology you are using to justify each step.

Then a = 2k + 1, b = __________, and c = 2k + 5 for some integer k. Suppose a2 + b2 = c2. Then (2k + 1)2 + (_________)2 = (2k + 5)2. It follows that 8k2 + 16k + 10 = 4k2 + 20k + 25 and 2 4k – 4k – _____ = 0. Thus k = 5/2 or k = _____. This contradicts k being an _______________. Therefore, there do not exist three consecutive odd integers a, b, and c such that a2 + b2 = c2. 12 best describes the structure of the proof? 25. Prove or give a counterexample: The sum of any five consecutive integers is divisible by five.

Let A = {2, 4, 6, 8}, B = {6, 7, 8, 9}, and C = {2, 8}. Which of the following statements are true? (a) {8, 7} ⊆ B (b) {7} ⊆ B ∩ C (c) (A \B ) ∩ C = {2} (d) C \ A = ∅ (e) ∅ ∈ B (f ) A ∩ B ∩ C = 8 (g) B \ A = {2, 4} (h) (B ∪ C) \ A = {7, 9} 4. Let A = {2, 4, 6, 8}, B = {6, 8, 10}, and C = {5, 6, 7, 8}. Find the following sets. (a) A ∩ B (b) A ∪ B (c) A \B (d) B ∩ C (e) B \C (f ) (B ∪ C )\A (g) (A ∩ B )\C (h) C \(A ∪ B) 5. Use Venn diagrams with three overlapping circles to illustrate each identity.

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

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