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By Steven R. Lay

Research with an creation to evidence, 5th version is helping fill within the foundation scholars have to achieve actual analysis-often thought of the main tricky direction within the undergraduate curriculum. through introducing good judgment and emphasizing the constitution and nature of the arguments used, this article is helping scholars flow conscientiously from computationally orientated classes to summary arithmetic with its emphasis on proofs. transparent expositions and examples, important perform difficulties, a variety of drawings, and chosen hints/answers make this article readable, student-oriented, and instructor- pleasant.   1. common sense and facts 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 keywords Index

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There are times, however, when order is important. For example, in analytic geometry the coordinates of a point (x, y) represent an ordered pair of numbers. The point (1, 3) is different from the point (3, 1). When we wish to indicate that a set of two elements a and b is ordered, we enclose the elements in parentheses: (a, b). † The important property of ordered pairs is that (a, b) = (c, d ) iff a = c and b = d. So far we have not really told you what ordered pairs are; we have only identified the property that they must satisfy.

Similar variations of the notation apply to intersections. 49 Sets and Functions There are some situations where a family of sets has not been indexed but we still wish to take the union or intersection of all the sets. If b is a nonempty collection of sets, then we let U B = {x : x ∈ B for some B ∈ b } B ∈b and I B = {x : x ∈ B for all B ∈ b }. 17 EXAMPLE For each k ∈ N, let Ak = [0, 2 – 1/k]. 18 PRACTICE Let S = {x ∈ ∞ U k =1 Ak = [0, 2). : x > 0}. For each x ∈ S, let Ax = (−1/x, 1/x). Find I Ax .

These operations are called union, intersection, and complementation. Intuitively, union may be thought of as putting together, intersection is like cutting down, and complementation corresponds to throwing out. 8 DEFINITION Let A and B be sets. The union of A and B (denoted A ∪ B), the intersection of A and B (denoted A ∩ B), and the complement of B in A (denoted A \ B) are given by A ∪ B = {x : x ∈ A or x ∈ B} A ∩ B = {x : x ∈ A and x ∈ B} A \ B = {x : x ∈ A and x ∉ B}. If A ∩ B = ∅, then A and B are said to be disjoint.

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