theory and problems of advanced calculus second edition - wrede & spiege

[...]... 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;... Problems 32 and 33) GEOMETRIC REPRESENTATION OF REAL NUMBERS The geometric representation of real numbers as points on a line called the real axis, as in the figure below, is also well known to the student For each real number there corresponds one and only one point on the line and conversely, i.e., there is a one-to-one (see Fig 1-1 ) correspondence between the set of real numbers and the set of points... If some elements of A and B are the same, we count them only once as in Problem 1.17 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... called the cardinality of the continuuum NEIGHBORHOODS The set of all points x such that jx À aj <  where  > 0, is called a  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... 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... > 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)... 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... a one-to-one correspondence between all rational numbers > 1 and all rational numbers in ð0; 1Þ Since these last are countable by Problem 1.17, it follows 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... 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 . strength, and no doubt the reason for the longevity of Professor Spiegel’s advanced calculus. His collection of solved and unsolved problems remains a part of this second edition. Advanced calculus. Either a > b, a ¼ b or a < b Law of trichotomy 2. If a > b and b > c, then a > c Law of transitivity 3. If a > b, then a þ c > b þ c 4. If a > b and c > 0, then ac >. class="bi x0 y0 w0 h0" alt="" 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