By Steven R. Lay

Research with an advent to facts, 5th variation is helping fill within the basis scholars have to reach actual analysis-often thought of the main tough path within the undergraduate curriculum. by way 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, various drawings, and chosen hints/answers make this article readable, student-oriented, and instructor- pleasant. 1. good judgment and evidence 2. units and capabilities three. the true Numbers four. Sequences five. Limits and Continuity 6. Differentiation 7. Integration eight. countless sequence Steven R. Lay thesaurus of key words Index

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**Example text**

E) For every positive integer n, n2 + n + 41 is prime. ( f ) Every prime is an odd number. (g) No integer greater than 100 is prime. (h ) 3n + 2 is prime for all positive integers n. ( i ) For every integer n > 3, 3n is divisible by 6. ( j ) If x and y are unequal positive integers and xy is a perfect square, then x and y are perfect squares. (k) For every real number x, there exists a real number y such that xy = 2. ( l ) The reciprocal of a real number x ≥ 1 is a real number y such that 0 < y < 1.

We have already seen the need for set notation in describing the context in which quantified statements are understood to apply. It is not our intention to develop set theory in a formal axiomatic way, but rather to discuss informally those aspects of set theory that are relevant to the study of analysis. In Section 1 we establish the basic notation for working with sets, and in the following two sections we apply this to the development of relations and functions. In Section 4 we compare the size of sets, giving special attention to infinite sets.

A) (A ∪ B) ∪ (U \A) (b) (A ∩ B) ∩ (U \A) (c) A ∩ [B ∪ (U \A)] (d) A ∪ [B ∩ (U \A)] (e) (A ∪ B) ∩ [A ∪ (U \B)] (f ) (A ∩ B) ∪ [A ∩ (U \B)] 7. Let A and B be subsets of a universal set U. Define the symmetric difference A B by A B = (A \B) ∪ (B \A). (a) Draw a Venn diagram for A B. (b) What is A A? (c) What is A ∅? (d) What is A U ? Sets and Functions 8. Let S = {∅, {∅}}. Determine whether each of the following is True or False. Explain your answers. (a) ∅ ⊆ S (b) ∅ ∈ S (c) {∅} ⊆ S (d) {∅} ∈ S 9.