Math 330 - 03 Homework (Fall 2018)

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Problem Set 11 (complete) Due: 11/12/2018. Board presentation: 11/16/2018

  1. Prove the following corollary to Prop. 10.4
    Corollary: $\glb(\R^+)=0$.
  2. Prove Prop. 10.7
  3. Prove Prop. 10.10.iii
  4. Prove Prop. 10.13.ii

Problem Set 10 (complete) Due: 11/05/2018. Board presentation: 11/14/2018

  1. Let $f:A\to B$ and $g:B\to C$ be functions.
    1. If $g\circ f$ is injective, then $f$ is injective.
    2. If $g\circ f$ is surjective, then $g$ is surjective.
  2. Construct examples of functions $f:A\to B$ and $g:B\to C$ such that:
    1. $g\circ f$ is injective, but $g$ is not injective.
    2. $g\circ f$ is surjective, but $f$ is not surjective.
  3. Prove Prop. 9.15 (Hint: induction)
  4. Prove Prop. 9.18

Problem Set 09 (complete) Due: 10/29/2018. Board presentation: 11/05/2018

  1. Prove Prop. 8.40.ii
  2. Prove Prop. 8.41
  3. Prove Prop. 8.50
  4. Give examples of subsets of $\R$ which are:
    1. bounded below and above,
    2. bounded below, but not bounded above,
    3. bounded above, but not bounded below,
    4. not bounded above or below.

Problem Set 08 (complete) Due: 10/22/2018. Board presentation: 10/31/2018

  1. Prove Prop. 6.16
  2. Prove Prop. 6.17
  3. Prove Prop. 6.25 (first part)
  4. Use Euclid's Lemma to prove the following corollary. Let $p$ be a prime, $k\in\N$, $m_1,m_2,\dots,m_k\in\N$. If $p|(m_1m_2\cdots m_k)$ then there is some $i$ with $1\leq i \leq k$ such that $p|m_i$. (Hint: Use induction on $k$).

Problem Set 07 (complete) Due: 10/15/2018. Board presentation: 10/31/2018

  1. Let $A$ be a set, and $\sim$ an equivalence relation on $A$. Let $A/\sim$ be the partition consisting of all equivalence classes of $\sim$. Let $\Theta_{(A/\sim)}$ be the equivalence relation induced by the partition $A/\sim$. Prove that $\Theta_{(A/\sim)}=\ \sim$.
  2. Do Project 6.8.iv.

Problem Set 06 (complete) Due: 10/08/2018. Board presentation: 10/31/2018

  1. Prove that set union is associative.
  2. Show, by counterexample, that set difference is not associative.
  3. Prove Prop. 5.20.ii
  4. Let $X$ and $Y$ be sets. Let $\power(X)$ denote the power set of $X$. Prove that: \[X\subseteq Y \iff \power(X)\subseteq\power(Y).\]
  5. (challenge) Prove that symmetric difference is associative.

Problem Set 05 (complete) Due: 10/01/2018. Board presentation: 10/05/2018

  1. Prove Prop. 4.6.iii
  2. Prove Prop. 4.11.ii
  3. Prove Prop. 4.15.i
  4. Prove Prop. 4.16.ii

Problem Set 04 (complete) Due: 09/17/2018. Board presentation: 09/21/2018

  1. Prove Prop. 2.38 (appendix)
  2. Prove Prop. 2.41.iii (appendix)

Problem Set 03 (complete) Due: 09/12/2018. Board presentation: 09/17/2018

  1. Prove that for all $k\in\N$, $k^2+k$ is divisible by 2.
  2. Prove Prop. 2.18.iii
  3. Prove Prop. 2.21. Hint: use proof by contradiction.
  4. Prove Prop. 2.23. Show, by counterexample, that the statement is not true if the hypothesis $m,n\in\N$ is removed.
  5. Fill-in the blank and prove that for all $k\geq\underline{\ \ }$, $k^2 < 2^k$.

Problem Set 02 (complete) Due:09/05/2018. Board presentation: 09/10/2018

  1. Prove Prop. 1.24
  2. Prove Prop. 1.27.ii,iv
  3. Prove Prop. 2.7.i,ii
  4. Prove Prop. 2.12.iii

Problem Set 01 (complete) Due: 08/27/2018. Board presentation: 08/31/2018

  1. Prove Prop. 1.7
  2. Prove Prop. 1.11.iv
  3. Prove Prop. 1.14
  4. Prove that 1 + 1 ≠ 1. (Hint: assume otherwise, and get a contradiction).
    Can you prove that 1 + 1 ≠ 0?

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