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Math 330 - 01 Homework (Spring 2022)


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Problem Set 13 (complete) Due: 05/09/2022

  1. Let f:A\to B and g:C\to D be functions. Define f\times g:A\times C \to B\times D by (f\times g)(a,c)=(f(a),g(c)).
    Prove that if f and g are surjective, then so is f\times g.
  2. Prove that the function \ f:\Z \to \N given by f(m) = \cases {2m &if $m>0,$ \cr -2m+1 &if $m\leq 0,$ \cr} is bijective.
  3. Prove that if A and B are finite sets, then so is A\union B. Morevoer, if A and B are disjoint, then |A\union B|=|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,w\in\R with z\neq 0\neq w, \frac{x}{z}+\frac{y}{w}=\frac{xw+yz}{zw}\qquad\textrm{and}\qquad\frac{x}{z}\frac{y}{w}=\frac{xy}{zw}
  3. Consider the set A=\{x\in\Q\mid x^2<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 \sqrt{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 x\in\R and r\in\R^+,
    (iv) |x| \leq r iff x \leq r and -x \leq r.
    (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:A\to B and g:B\to C be functions.
    1. Prove Prop. 9.7.ii
    2. Prove that if g\circ f 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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