1 Exercises for Unit I (General considerations) I.1 : Overview of the course (Halmos, Preface; Lipschutz, Preface) Questions to answer: 1. Why are there safety codes mandating that the foundation of a building must meet certain structural requirements? How might one apply the same principle to the mathematical sciences? 2. Why do manufacturers invest substantial resources into studying their mass production methods? How might one apply the same principle to the mathematical sciences? 3. Emphasis on quality standards can sometimes be taken too far. For example, in a chemical laboratory one could focus on cleaning laboratory equipment to a point that it would interfere with performing the experiments that are supposed to be carried out. How might one apply the same principle to the mathematical sciences? 4. The notes mentioned that one reason for the continued study of axiomatic set theory is to test the limits to which the foundations of mathematics can be pushed. Give one or more examples outside of the mathematical sciences where manufacturers might be expected to test the limits of their products. I.2 : Historical background and motivation Questions to answer: 1. Zeno's paradoxes are based on mixing "atomic" and "continuous" models for physical phenomena too casually. The following example shows that the casual use of mixed models still happens today: Consider the problem of finding the center of mass for an object like a solid hemisphere by means of calculus. Using calculus to find the center of mass tacitly assumes that matter is continuous. On the other hand, the atomic theory of matter implies that matter is not infinitely divisible. In view of this discrepancy, what meaning should be attached to the integral formula for the center of mass for the solid hemisphere? 2. Find the gap in the following proof that the angle sum S of a triangle is always 180 degrees: Let A, B, C be the vertices of the triangle, and let D be a point between B and C so that ∆∆∆∆ ABC is split into ∆∆∆∆ ABD and ∆∆∆∆ ADC. The sums of the angles in both triangles are easily computed to be equal to the sums of the angles in ∆∆∆∆ ABC plus 1802 degrees. But since the angle sum of a triangle is S, this sum is also equal to 2 S and hence we have 2 S = S + 180, which implies that S must be equal to 180. Where is the mistake in this argument? — This issue is relevant to the discovery of non – Euclidean geometry because Euclid’s Fifth Postulate is equivalent to assuming that the angle sum of a triangle is always 180 degrees. [ Comment : It will probably be helpful to draw a picture corresponding to the assertions made in the fallacious proof. ] 3. The following example in coordinate geometry illustrates the need for adding assumptions to those in Euclid's Elements. Consider the triangle in the coordinate plane with vertices (0, 1), (0, 0) and (1, 0), and let L be a line passing through the midpoint of the hypotenuse, which is (½, ½). Show that L either goes through the vertex (0, 0) or else contains a point on one of the other two sides. It might be helpful to break this problem up into cases depending upon the slope m of the line, which might be equal to 1, greater than 1, undefined, less than – 1, or between – 1 and + 1. — This example reflects a general fact about triangles and lines meeting one of the sides between the two endpoints, and it is known as Pasch's Postulate after M. Pasch (1843 – 1930), who noted its significance. This statement is used in the Elements, but it is not proven and the need to assume it is not acknowledged. [Comment: This is not meant to denigrate the Elements; the purpose is to illustrate that the sorts of problems in Euclid's important and monumental work that were discovered in the late 19th century.] 4. Two ordinary decks of 52 cards are shuffled, and each of 52 players is dealt one card from each deck. Explain why at least half of the players will receive two number cards (Ace through 10). [Hint: It might be good to break things down into four cases, depending on what sort of card an individual receives from each deck. The number of persons receiving either a number or picture card from a fixed is of course 52, and the number of persons receiving a number card from the first or second deck is 40. Of course, the number of persons in each of the four cases is also nonnegative.] 5. The following example shows that some care must be taken when rearranging the terms of an infinite series because different arrangements of the terms sometimes lead to different answers. Consider the standard infinite series for the natural logarithm of 2 and suppose we rearrange the terms as follows: ...1211015181613141211 +−−+−−+−− Explain why the sum of the rearranged series is ½ ln 2 . — The general rearragement question is discussed in pages 75 – 78 of Rudin, Principles of Mathematical Analysis (3rd Ed., McGraw – Hill, New York, 1976, ISBN: 0–07–054235–X); the subsequent proof of Theorem 3.50 is also relevant to this topic, and additional examples involving the effect of rearrangement on infinite series appear on pages 73 – 74 of Rudin. There is also a discussion with examples on pages 656 – 657 and 660 of the 10th Edition of3 Thomas’ Calculus, for which bibliographic information is given in the notes. Two points especially worth noting are that the sum of a series of nonnegative terms does not change if one rearranges the terms in any manner whatsoever and the sum of any series does not change if we only rearrange finitely many terms. More generally, the sum does not change if the series is absolutely convergent (i.e., the series whose terms are the absolute values of the given ones converges). Of course, the latter fails for the series considered above. 6. One important fact about power series is that they can be differentiated term by term, and the result will be the derivative of the function represented by the series. The following example shows that term–by–term differentiation of trigonometric series is not possible. Consider the expansion for the square wave function described in the notes, and consider the termwise second derivatives of both sides away from the points of discontinuity. If it were possible to perform term–by–term differentiation on this