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Math 330 - 01 Homework (Spring 2022)

  • 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 13 (complete) Due: 05/09/2022

  1. Let f:AB and g:CD be functions. Define f×g:A×CB×D by (f×g)(a,c)=(f(a),g(c)).
    Prove that if f and g are surjective, then so is f×g.
  2. Prove that the function  f:ZN given by f(m)={2mif m>0, 2m+1if m0,  is bijective.
  3. Prove that if A and B are finite sets, then so is AB. Morevoer, if A and B are disjoint, then |AB|=|A|+|B|.
  4. Prove Theorem 13.28. Hint: consider the function tan(x) from calculus.

Problem Set 12 (complete) Due: 05/02/2022. Board presentation: 05/06/2022

  1. Prove the converse of Prop 11.2
  2. Prove that for all x,y,z,wR with z0w, xz+yw=xw+yzzwandxzyw=xyzw
  3. 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.
  4. Prove Prop. 11.21.iii

Problem Set 11 (complete) Due: 04/19/2022. Board presentation: 04/22/2022

  1. Prove part (iv) of lemma stated in class:
    for xR and rR+,
    (iv) |x|r iff xr and xr.
    (Hint: use part (iii) of the same lemma.
  2. Prove Prop. 10.10.iii (Hint: use 10.8.iv)
  3. Prove Prop. 10.13.ii
  4. Prove Prop. 10.17 (Hint: use induction)

Problem Set 10 (complete) Due: 04/11/2022. Board presentation: 04/15/2022

  1. Let f:AB and g:BC be functions.
    1. Prove Prop. 9.7.ii
    2. Prove that if gf is surjective, then g is surjective.
  2. Prove Prop. 9.10.ii
  3. Prove Prop. 9.15 (Hint: induction)
  4. Prove Prop. 9.18

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people/fer/330ws/spring2022/homework.txt · Last modified: 2022/05/06 06:51 by fer