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Math 330 - 02 Homework (Fall 2017)

  • LaTeX-ed solutions are encouraged and appreciated.
  • If you use LaTeX, hand-in a printed version of your homework.
  • You are encouraged to discuss homework problems with classmates, but such discussions should NOT include the exchange of any written material.
  • Writing of homework problems should be done on an individual basis.
  • References to results from the textbook and/or class notes should be included.
  • The following lists should be considered partial and tentative lists until the word complete appears next to it.
  • Use 8.5in x 11in paper with smooth borders. Write your name on top of each page. Staple all pages.

Problem Set 12 (complete) Due: 12/08/2017. Board presentation: 12/08/2017

  1. Prove that if A and B are finite sets, then AB is a finite set.
  2. Prove the following corollary to Proposition 13.6.
    1. If f:AB is injective and B is finite, then A is finite.
    2. If g:AB is surjective and A is finite, then B is finite.
  3. Do Project 13.15, finding a formula for the bijection in the picture.
  4. Prove Theorem 13.28.

Problem Set 11 (complete) Due: 12/01/2017. Board Presentation: 12/01/2017

  1. Write down the details of the proofs that the sum of a rational number and an irrational number is irrational, and that the product of a non-zero rational number and an irrational number is irrational.
  2. Prove the converse of Prop. 11.2
  3. Do Project 11.14
  4. Prove that for all x,y,z,wR with z,w0, xz+yw=xw+yzzwandxzyw=xyzw
  5. Consider the set A={xQx2<2} Show that A is non-empty and has an upper bound in Q, but does not have a least upper bound in Q. Hint: by way of contradiction, assume A has a least upper bound u in Q, and compare it with 2.
  6. Consider the sequence defined recursively by an=an1+3an2a1=1a2=2. Use the converse of Proposition 11.25 to find a closed formula for an.

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people/fer/330ws/fall2017/homework.txt · Last modified: 2018/08/24 09:05 by fer