#let m(math) = align(center)[$#math$]
#let pgbreakmsg = align(center, text(blue, weight: "black", size: 1.5em)[See Next Page\ ↓])

#let solve(work, solution) = align(
  center,
)[
  #let solution = align(center, block(
    inset: 5pt,
    stroke: blue + .3pt,
    fill: rgb(0, 149, 255, 15%),
    radius: 4pt,
  )[#align(left)[#solution]])

  #if work == [] [
    #solution
  ] else [
    #block(inset: 6pt, radius: 4pt, stroke: luma(50%) + .5pt, fill: luma(90%))[
      #align(left, text(font: "Liberation Sans", size: .85em, work))
      #solution
    ]
  ]
]

#let problem-header(number, points) = [== Problem #number. #text(weight: "regular")[[#points
    points]]]

#let problem(number, points, body) = [
  == Problem #number. #text(weight: "regular")[[#points points]]
  #body
]

#set page(margin: (x: .4in, y: .4in))
#set table(align: center)

_*Price Hiller*_
#v(-.8em)
_*zfp106*_
#v(-.8em)
Homework Assignment 3
#v(-.8em)
CS 2233
#v(-.8em)
Section 001

#align(
  center,
  block(
    inset: 6pt,
    radius: 4pt,
    stroke: luma(50%) + .5pt,
    fill: luma(90%),
  )[If you are interested in viewing the source code of this document, you can do so
    by clicking
    #text(
      blue,
      link(
        "https://git.orion-technologies.io/Price/college/src/branch/Development/Spring-2023/CS-2233/Assignment-3/Solution.typ",
        "here",
      ),
    ).],
)
= Problems

#problem(
  1,
  10,
)[
  - Complete all participation activities in zyBook sections $2.1$, $2.2$, $2.4$-$2.6$.
  #solve[][Done]
]

#problem(
  2,
  10,
)[
  Prove that if $a$, $b$, and $c$ are odd integers, then $a + b + c$ is an odd
  integer.

  #solve[
    An odd integer is expressed as $2k + 1$ where $k$ is some integer.
  ][
    $a + b + c$ can be rewritten as $(2z + 1) + (2n + 1) + (2p + 1)$. Working this
    equation we end up with: $2z + 2n + 2p + 3$. We can then factor out $2$ giving
    us $2(z + n + p) + 3$ and with some additional manipulation we can get $2((z + n + p) + 1) + 1$.
    Notice the inner part ($(z + n
    + p) + 1 $) could be re-expressed as $k$, thus we can rexpress the entire thing
    (with a substitution) as $2k + 1$ where $k = (z + n + p + 1)$. Therefore, if $a$, $b$,
    and $c$ are odd integers, then $a + b + c$ is an odd integer.
  ]
]
#problem(
  3,
  30,
)[
  Recall that a rational number can be put in the form $p/q$ where $p$ and $q$ are
  integers and $q
  ≠ 0$ Prove the following for any rational number, $x$:

  a.) If $x$ is rational, then $x - 5$ is rational

  b.) If $x - 5$ is rational, then $x/3$ is rational

  c.) If $x/3$ is rational, then $x$ is rational
]

#pgbreakmsg
#pagebreak()
#problem(
  4,
  20,
)[
  Consider the following statement: For all integers $m$ and $n$, if $m - n$ is
  odd, then $m$ is odd or $n$ is odd.

  a. Prove the statement using a proof by contrapositive

  b. Prove the statement by using a proof by contradiction
]