75 lines
2.6 KiB
Plaintext
75 lines
2.6 KiB
Plaintext
#set page(margin: (x: .5in, y: .5in))
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#let solve(solution) = [
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#let solution = align(
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center,
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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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)
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#solution
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]
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#align(center)[
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= CS 3333 Mathematical Foundations
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Homework 1 (100 points)\
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#underline[Price Hiller] *|* #underline[zfp106]
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]
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*Submission:*
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+ Submit a single PDF (not Word) file through Canvas.
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+ You can either edit the Word file directly or write down your solution in any other text document using other editors.
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+ Convert the Word document or the text document into a single PDF file.
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+ If there is any plagiarism, you will lose all points on the questions at first time. In next, you will lose all points in the whole homework.
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#line(length: 100%, stroke: .25pt)
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*Questions:*
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1. Does 41 divide each of these numbers (just answer yes or no)? (16 points)
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#solve[#align(center)[]#grid(
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columns: (100pt, 100pt),
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rows: (10pt, auto),
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gutter: 3pt,
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[a) 123 *(yes)*], [b) 92 *(no)*],
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[c) 413 *(no)*], [d) 1640 *(yes)*],
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)]
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2. Prove/Show that if $a | b$ and $b | a$, where $a$ and $b$ are integers, then $a = b$ or $a = -b$. (20 points)
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#solve[
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+ $b = a k$, where $k$ is an integer
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+ $a = b q$, where $q$ is an integer
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+ Use $a = b q$
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+ Substitute $b$: $a = a k q$
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+ Algebra: $k q = 1$
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+ ∵ $k$ & $q$ are integers, $k$ and $q$ must be $1$ or $-1$
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+ ∴ $a = b$ or $a = -b$
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]
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3. Prove that if $a$, $b$, $c$, and $d$ are integers, where $a ≠ 0$ and $b ≠ 0$, such that $a | c$ and $b | d$, then $a b | c d$. (20 points)
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#solve[
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+ $c = k a$, where $k$ is an integer
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+ $d = q b$, where $q$ is an integer
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+ $c d = (k a)(q b) = (k q)(a b)$
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+ $a b | (k q)(a b)$ is true
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+ ∴ $a b | c d$
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]
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4. When $x$ is an integer, prove that $1 + x^2 <= (1 + x)^2$ is true or disprove it by providing a counterexample. (20 points)
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#solve[
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+ False. Counterexample: let $x = -1$
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+ Substitute $x$: $1 + (-1)^2 <= (1 + (-1))^2$
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+ Solve: $2 <= 0$, not true
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+ ∴ when $x$ is an integer $1 + x^2 <= (1 + x)^2$ is false
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]
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5. Prove that if $a$ is an integer that is not divisible by $3$, denoted as $3 ∤ a$, then $(a + 1)(a + 2)$ is divisible by $3$, denoted as $3 | (a + 1)(a + 2)$. (24 points)
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#solve[
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+ Prove: $3 | (a + 1)(a + 2)$
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+ Assume $a = 3k + 1$, where $k$ is an integer
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+ Substitute $a$: $((3k + 1) + 1)((3k + 1) + 2)$
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+ Algebra: $9k^2 + 15k + 6$
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+ Factor $3$: $3(3k^2 + 5k + 2)$
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+ $3$ times some integer is necessarily divisible by $3$ according to divisibility
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+ ∴ If $3 ∤ a$, then $3 | (a + 1)(a + 2)$
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]
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