175 – Quiz 2

September 19, 2009

Here is quiz 2.

Problem 1 is (simplification of part of) exercise 6.6.16 from the book.

To solve the question, use coordinates as in the accompanying figure in the book, so the origin is at ground level, and y increases downwards. The units of y are feet. For a fix y with 0\le y\le 20, the thin slice of water in the tank at depth y and of tickness dy has volume dV=10\times 12\times dy and weighs dF=62.4\times dV=7488\,dy. This is a constant force, so the work required to remove it to ground level is just dW={\rm distance}\times dF, where {\rm distance} is the depth at which the slice is located, i.e., y. Hence, dW=7488 y\,dy. The total work is obtained by adding all these contributions, i.e., W=\int_0^{20} 7488 y\,dy=7488\times 200=1497600 ft-lb.

Problem 2 is exercise 9.2.16 from the book. 

The curve is r^2=-\cos\theta. Since \cos\theta=\cos(-\theta), the graph is symmetric about the x-axis (because whenever (r,\theta) is in the graph, then so is (r,-\theta)).

Since (-r)^2=r^2, the graph is symmetric about the origin (because whenever (r,\theta) is in the graph, then so is (-r,\theta)).

Since the graph is symmetric about both the origin and the x-axis, it is also symmetric about the y-axis.

To sketch the curve, look first at 0\le \theta< \pi/2. Here \cos\theta>0, so r^2<0, which is impossible, so there is nothing to graph here. Consider now what happens when \pi/2\le\theta\le \pi. As \theta increases, -\cos\theta increases, from 0 to 1. So the same occurs with r^2. This means that r increases from 0 to 1, and -r decreases from 0 to -1. The part with r gives us a curve in the second quadrant, and the part with -r gives us its reflection about the origin. This part of the curve is in the fourth quadrant. Their reflections on the x-axis complete the curve, which can be seen here.

Note that \tan(\pi/2) is undefined. This corresponds to the fact that at the origin the tangent to the curve is the y-axis, as can be seen from the graph.

502 – Predicate logic

September 19, 2009

These exercises (due September 28) are mostly meant to test your understanding of compactness.

  1. Let M be a nonstandard model of {\rm Th}({\mathbb N},+,\times,0,1,<). Show:
    1. (Overspill) Suppose that A\subset M is definable (with parameters) and that A\subseteq{\mathbb N}. Show that A is finite.
    2. (Underspill) Suppose that A\subset M is definable and that A\cap {\mathbb N}=\emptyset. Show that there is some infinite c such that all the elements of A are larger than c.
  2. Let M be a nonstandard model of {\rm Th}({\mathbb R},{\mathbb N},+,\times,<,\dots). Here, {\mathbb N} is treated as a relation, and in \dots we may have placed whatever functions and relations we may have need to reference in what follows; moreover, we assume that in our language we have a constant symbol for each real number. (Of course, this means that we are lifting the restriction that languages are countable.) To ease notation, let’s write ({}^*{\mathbb R},{}^*{\mathbb N},{}^*+,{}^*\times,{}^*<,\dots) for M. The convention is that we identify actual reals in {\mathbb R} with their copies in {}^*{\mathbb R}, so we write \pi rather than {}^*\pi, etc.
    1. Show that ({}^*{\mathbb N},{}^*+\upharpoonright {}^*{\mathbb N}\times {}^*{\mathbb N},{}^*\times\upharpoonright {}^*{\mathbb N}\times {}^*{\mathbb N},0,1,{}^*<\upharpoonright {}^*{\mathbb N}\times {}^*{\mathbb N}) is a nonstandard model of the theory of problem 1. (In particular, check that the indicated restrictions of {}^*+ and {}^*\times have range contained in {}^*{\mathbb N}.)
    2. A (nonstandard) real r is finite iff there is some (finite) natural number n such that -n<r<n. Otherwise, it is infinite. A (nonstandard) real r is infinitesimal iff r\ne 0 but for all positive (finite) natural numbers n, one has that -1/n<r<1/n. We write r\approx 0 to mean that either r is infinitesimal, or else it is 0. Show that infinite and infinitesimal numbers exist. The monad of a real r is the set of all s such that r-s\approx 0, which we may also write as r\approx s; and say that r and s are infinitesimally close. Show that the relation \approx is an equivalence relation. Show that if a monad contains an actual real number, then this number is unique. Show that this is the case precisely if it is the monad of a finite number. In this case, write s={\rm st}(r) to indicate that the (actual) real s is in the monad of r. We also say that s is the standard part of r.
    3. Show that a function f is continuous at a real a iff {\rm st}({}^*f(a+h))=f(a) for all infinitesimal numbers h.
    4. Suppose that f is continuous on the closed interval {}[0,1]. Argue as follows to show that f attains its maximum: For each positive integer n, there is some integer i with 0\le i\le n such that f(i/n)={\rm max}\{f(j/n)\mid 0\le j\le n\}. Conclude that the same holds if n is some infinite natural number, i.e., there is some (perhaps infinite) “natural number” i with 0\le i\le n such that {}^*f(i/n)={\rm max}\{{}^*f(j/n)\mid 0\le j\le n\}. Let r={\rm st}(i/n), and argue that the maximum of f is attained at r.