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By Steven R. Lay

Research with an advent to facts, 5th version is helping fill within the basis scholars have to reach actual analysis-often thought of the main tough 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 flow 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 facts 2. units and services three. the genuine Numbers four. Sequences five. Limits and Continuity 6. Differentiation 7. Integration eight. endless sequence Steven R. Lay thesaurus of keyword phrases Index

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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. 13(a). THEOREM: Let A be a subset of U. Then A ∪ (U \A) = U. Proof: If x ∈ A ∪ (U \A), then x ∈ __________ or x ∈ __________. Since both A and U \A are subsets of U, in either case we have __________. Thus ____________ ⊆ ____________. On the other hand, suppose that x ∈ ____________.

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. 9 PRACTICE iff iff iff (x ∈ A) ∨ (x ∈ B) (x ∈ A) ∧ (x ∈ B) (x ∈ A) ∧ ~ (x ∈ B).

If x ∈ A, then __________ ________________________________________________________. Thus A ⊆ B. ♦ 12. Suppose you are to prove that set A is a subset of set B. Write a reasonable beginning sentence for the proof, and indicate what you would have to show in order to finish the proof. 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?

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