Schaum''s outline of theory and problems of advanced calculus

[...]... Then the set of elements belonging to A or B (or both) is countable The set consisting of all elements which belong to A or B (or both) is often called the union of A and B, denoted by A [ B or A þ B The set consisting of all elements which are contained in both A and B is called the intersection of A and B, denoted by A \ B or AB If A and B are countable, so is A \ B " The set consisting of all elements... neighborhood of the point a The set of all points x such that 0 < jx À aj <  in which x ¼ a is excluded, is called a deleted  neighborhood of a or an open ball of radius  about a LIMIT POINTS A limit point, point of accumulation, or cluster point of a set of numbers is a  number l such that every deleted  neighborhood of l contains members of the set; that is, no matter how small the radius of a ball... on the line Because of this we often use point and number interchangeably _4 _p _5 _4 1 2 3 _3 _2 _1 0 1 √2 2 e 3 p 4 5 Fig 1-1 (The interchangeability of point and number is by no means self-evident; in fact, axioms supporting the relation of geometry and numbers are necessary The Cantor–Dedekind Theorem is fundamental.) The set of real numbers to the right of 0 is called the set of positive numbers;... a subset of the set of complex numbers with b ¼ 0 The complex number 0 þ 0i corresponds to the real number 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi The absolute value or modulus of a þ bi is defined as ja þ bij ¼ a2 þ b2 The complex conjugate of " a þ bi is defined as a À bi The complex conjugate of the complex number z is often indicated by z or zà The set of complex numbers obeys rules 1 through 9 of Page 2, and thus... This is part of the proof establishing the induction and may be difficult or impossible 4 Since the statement is true for n ¼ 1 [from step 1] it must [from step 3] be true for n ¼ 1 þ 1 ¼ 2 and from this for n ¼ 2 þ 1 ¼ 3, and so on, and so must be true for all positive integers (This assumption, which provides the link for the truth of a statement for a finite number of cases to the truth of that statement... left of 0 is the set of negative numbers, while 0 itself is neither positive nor negative (Both the horizontal position of the line and the placement of positive and negative numbers to the right and left, respectively, are conventions.) Between any two rational numbers (or irrational numbers) on the line there are infinitely many rational (and irrational) numbers This leads us to call the set of rational... correspondence between the points of the interval 0 @ x @ 1 and À5 @ x @ À 3 (b) What is the cardinal number of the sets in (a)? Ans (b) C, the cardinal number of the continuum 1.59 (a) Prove that the set of all rational numbers is countable (b) What is the cardinal number of the set in (a)? Ans (b) Fo 1.60 Prove that the set of (a) all real numbers, 1.61 The intersection of two sets A and B, denoted by A \ B... > 0 and x þ 3 < 0, i.e., x > 4 and x < À3 This is impossible, since x cannot be both greater than 4 and less than À3 Case 2: x À 4 < 0 and x þ 3 > 0, i.e x < 4 and x > À3 This is possible when À3 < x < 4 inequality holds for the set of all x such that À3 < x < 4 1.12 If a A 0 and b A 0, prove that 1 ða þ bÞ A 2 Thus the pffiffiffiffiffi ab The statement is self-evident in the following cases (1) a ¼ b, and (2)... real numbers, then: 1 Either a > b, a ¼ b or a < b 2 If a > b and b > c, then a > c Law of trichotomy Law of transitivity 3 If a > b, then a þ c > b þ c 4 If a > b and c > 0, then ac > bc 5 If a > b and c < 0, then ac < bc ABSOLUTE VALUE OF REAL NUMBERS The absolute value of a real number a, denoted by jaj, is defined as a if a > 0, Àa if a < 0, and 0 if a ¼ 0 EXAMPLES j À 5j ¼ 5, j þ 2j ¼ 2, j À 3 j ¼... that the set of all rational numbers > 1 is also countable From Problem 1.18 it then follows that the set consisting of all positive rational numbers is countable, since this is composed of the two countable sets of rationals between 0 and 1 and those greater than or equal to 1 From this we can show that the set of all rational numbers is countable (see Problem 1.59) 1.20 Prove that the set of all real . Theory and Problems of ADVANCED CALCULUS Second Edition ROBERT WREDE, Ph.D. MURRAY R. SPIEGEL, Ph.D. Former Professor and Chairman of Mathematics Rensselaer Polytechnic Institute Hartford Graduate. appears in the print version of this title: 0-0 7-1 3756 7-8 All trademarks are trademarks of their respective owners. Rather than put a trademark symbol after every occurrence of a trademarked. uninterrupted or error free. Neither McGraw-Hill nor its licensors shall be liable to you or anyone else for any inaccuracy, error or omission, regardless of cause, in the work or for any dam- ages