Homework


Homework Paper
  • With lines.
  • Without lines.

    • Homework 1 - Due: Feb 2; Final Due Date: 15 Feb
  • 1.8, 1.11(ii), 1.11(v).

    • Homework 2 - Due: Feb 3; Final Due Date: 16 Feb
  • 1.16.

    • Homework 3 - Due: Feb 6; Final Due Date: 17 Feb
  • 1.22, 1.25(i), 1.25(ii).

    • Homework 4 - Due: Feb 8; Final Due Date: 22 Feb
  • 1.27(ii).

    • Homework 5 - Due: Feb 9; Final Due Date: 22 Feb
  • 2.6, 2.7(ii), 2.8.

    • Homework 6 - Due: Feb 10; Final Due Date: 23 Feb
  • 2.10, 2.12(iii).

    • Homework 7 - Due: Feb 13; Final Due Date: 27 Feb
  • 2.18(iii), 2.20.

    • Hint for 2.18(iii). Use the following lemma I proved in class: Lemma. $\forall k\in \mathbb{N}$, $k^2+k$ is even.

    Homework 8 - Due: Feb 15; Final Due Date: 27 Feb
  • 2.23, 2.27.

    • Homework 9 - Due: Feb 16; Final Due Date: 29 Feb
  • 2.33.

    • Hint for 2.33. Consider the set $C:=\{a-b+1:a\in A\}$. Show that $C$ is a nonempty subset of $\mathbb{N}$. Then use the Well-Ordering Principle (Theorem 2.32) and use the smallest element of $C$ to find the smallest element of $A$.

    Homework 10 - Due: Feb 22; Final Due Date: 2 Mar
  • 4.5.

    • Homework 11 - Due: Feb 23; Final Due Date: 7 Mar
  • 4.7(i).

    • Homework 12 - Due: Feb 27; Final Due Date: 8 Mar
  • 4.13, 4.15(iii), 4.16(ii), 4.17.

    • Binomial Theorem Practice (You do not need to turn this in)
  • Find the last term of $(x+2y)^5$.
  • Find the last term of $(x-2y)^5$.
  • Find the third term of $(2x-3y)^9$.
  • Prove: $\forall k\in \mathbb{Z}_{\geq 0}$, $\displaystyle \sum_{j=0}^k \binom{k}{j}2^j=3^k$.
  • Prove: $\forall k\in \mathbb{Z}_{\geq 0}$, $\displaystyle \sum_{j=0}^k (-1)^j\binom{k}{j}=0$.
  • Prove: let $x\in \mathbb{Z}$, $\forall k\in \mathbb{Z}_{\geq 0}$, $\displaystyle \sum_{j=0}^k \binom{k}{j}x^j=(1+x)^k$.
  • Check out the Multinomial Theorem. The coefficients of the multinomial expansion can be realized by a generalized version of Pascal's Triangle to higher dimensions, called Pascal's simplex.

    • Homework 13 - Due: Mar 2; Final Due Date: 15 Mar
  • 4.30.

    • Homework 14 - Due: Mar 5; Final Due Date: 16 Mar
  • 5.3(iii), 5.7.

    • Homework 15 - Due: Mar 7; Final Due Date: 19 Mar
  • 5.15.

    • Hint for 5.15. Use the following:
  • $x\in A\cap B$ $\iff$ $x\in A$ and $x\in B$.
  • $x\in (A\cap B)^\text{c}$ $\iff$ $x\notin A\cap B$ $\iff$ $x\notin A$ or $x\notin B$. (Notice that this is the negation of: $x\in A$ and $x\in B$.)
  • $x\in A\cup B$ $\iff$ $x\in A$ or $x\in B$.
  • $x\in (A\cup B)^\text{c}$ $\iff$ $x\notin A\cup B$ $\iff$ $x\notin A$ and $x\notin B$. (Notice that this is the negation of: $x\in A$ or $x\in B$.)

    • Homework 16 - Due: Mar 8; Final Due Date: 21 Mar
  • 5.20.

    • Homework 17 - Due: Mar 12; Final Due Date: 23 Mar
  • 6.5.

    • Hint for 6.5. If $p$ and $q$ are statements, then to prove "$p$ OR $q$" you may use either of the following methods:
  • Method 1: suppose $p$ is false, and show $q$ must be true.
  • Method 2: suppose $q$ is false, and show $p$ must be true.
    • N.B. You do not need to use both methods (just pick one!)! Why do these methods work? Also, what are some other methods to prove "$p$ OR $q$"?

    Homework 18 - Due: Mar 16; 1st Version Final Due Date: Mar 21; Rewrite Final Due Date: 29 Mar
  • 6.15.

    • Homework 19 - Due: Mar 19; 1st Version Final Due Date: Mar 23; Rewrite Final Due Date: 30 Mar
  • 6.20.

    • Homework 20 - Due: Mar 26; 1st Version Final Due Date: Mar 29; Rewrite Final Due Date: 13 April
  • 6.25.

    • Homework 21 - Due: Mar 29; 1st Version Final Due Date: April 12; Rewrite Final Due Date: 18 April
  • 6.32, 6.35.

    • Homework 22 - Due: April 12; 1st Version Final Due Date: April 16; Rewrite Final Due Date: 20 April
  • 8.6, 8.40(ii), 8.43.

    • Homework 23 - Due: April 12; 1st Version Final Due Date: April 18; Rewrite Final Due Date: 25 April
  • 8.50.

    • Homework 24 - Due: April 18; 1st Version Final Due Date: April 20; Rewrite Final Due Date: 30 April
  • 9.11, 9.15.

    • Homework 25 - Due: April 19; 1st Version Final Due Date: April 23; Rewrite Final Due Date: 2 May
  • 9.18.

    • Homework 26 - Due: April 20; 1st Version Final Due Date: April 26; Rewrite Final Due Date: 3 May
  • 10.7.

    • Homework 27 - Due: April 23; 1st Version Final Due Date: April 27; Rewrite Final Due Date: 4 May
  • 10.9.

    • Homework 28 - Due: April 25; 1st Version Final Due Date: April 30; Rewrite Final Due Date: 7 May
  • 10.17.

    • Homework 29 - Due: April 26; 1st Version Final Due Date: May 2; Rewrite Final Due Date: 9 May
  • 10.21(i), 10.23(iii).

    • Homework 30 - Due: April 27; 1st Version Final Due Date: May 3; Rewrite Final Due Date: 10 May
  • 10.27, 10.28, 11.2, 11.6.

    • Homework 31 - Due: April 30; 1st Version Final Due Date: May 4; Rewrite Final Due Date: 11 May
  • 11.12.

    • Homework 32 - Due: May 2; 1st Version Final Due Date: May 4; Rewrite Final Due Date: 11 May
  • 11.17.

    • Homework 33 - Due: May 3; 1st Version Final Due Date: May 9; Rewrite Final Due Date: 11 May
  • 11.25.

    • Homework 34 - Due: May 7; 1st Version Final Due Date: May 10; Rewrite Final Due Date: 11 May
  • 13.2.

    • Homework 35 (Optional) - Due 11 May
  • 13.6.

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