119 lines
2.8 KiB
Plaintext
119 lines
2.8 KiB
Plaintext
#let m(math) = align(center)[$#math$]
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#let pgbreakmsg = align(center, text(blue, weight: "black", size: 1.5em)[See Next Page\ ↓])
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#let solve(work, solution) = align(
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center,
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)[
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#let solution = align(center, block(
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inset: 5pt,
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stroke: blue + .3pt,
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fill: rgb(0, 149, 255, 15%),
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radius: 4pt,
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)[#align(left)[#solution]])
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#if work == [] [
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#solution
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] else [
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#block(inset: 6pt, radius: 4pt, stroke: luma(50%) + .5pt, fill: luma(90%))[
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#align(left, text(font: "Liberation Sans", size: .85em, work))
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#solution
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]
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]
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]
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#let problem-header(number, points) = [== Problem #number. #text(weight: "regular")[[#points
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points]]]
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#let problem(number, points, body) = [
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== Problem #number. #text(weight: "regular")[[#points points]]
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#body
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]
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#set page(margin: (x: .4in, y: .4in))
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#set table(align: center)
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_*Price Hiller*_
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#v(-.8em)
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_*zfp106*_
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#v(-.8em)
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Homework Assignment 3
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#v(-.8em)
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CS 2233
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#v(-.8em)
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Section 001
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#align(
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center,
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block(
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inset: 6pt,
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radius: 4pt,
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stroke: luma(50%) + .5pt,
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fill: luma(90%),
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)[If you are interested in viewing the source code of this document, you can do so
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by clicking
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#text(
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blue,
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link(
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"https://git.orion-technologies.io/Price/college/src/branch/Development/Spring-2023/CS-2233/Assignment-3/Solution.typ",
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"here",
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),
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).],
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)
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= Problems
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#problem(
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1,
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10,
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)[
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- Complete all participation activities in zyBook sections $2.1$, $2.2$, $2.4$-$2.6$.
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#solve[][Done]
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]
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#problem(
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2,
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10,
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)[
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Prove that if $a$, $b$, and $c$ are odd integers, then $a + b + c$ is an odd
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integer.
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#solve[
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An odd integer is expressed as $2k + 1$ where $k$ is some integer.
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][
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$a + b + c$ can be rewritten as $(2z + 1) + (2n + 1) + (2p + 1)$. Working this
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equation we end up with: $2z + 2n + 2p + 3$. We can then factor out $2$ giving
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us $2(z + n + p) + 3$ and with some additional manipulation we can get $2((z + n + p) + 1) + 1$.
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Notice the inner part ($(z + n
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+ p) + 1 $) could be re-expressed as $k$, thus we can rexpress the entire thing
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(with a substitution) as $2k + 1$ where $k = (z + n + p + 1)$. Therefore, if $a$, $b$,
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and $c$ are odd integers, then $a + b + c$ is an odd integer.
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]
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]
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#problem(
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3,
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30,
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)[
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Recall that a rational number can be put in the form $p/q$ where $p$ and $q$ are
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integers and $q
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≠ 0$ Prove the following for any rational number, $x$:
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a.) If $x$ is rational, then $x - 5$ is rational
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b.) If $x - 5$ is rational, then $x/3$ is rational
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c.) If $x/3$ is rational, then $x$ is rational
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]
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#pgbreakmsg
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#pagebreak()
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#problem(
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4,
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20,
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)[
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Consider the following statement: For all integers $m$ and $n$, if $m - n$ is
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odd, then $m$ is odd or $n$ is odd.
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a. Prove the statement using a proof by contrapositive
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b. Prove the statement by using a proof by contradiction
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]
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