formal-homework:0.1.0

packages/preview/formal-homework/0.1.0/README.md

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+# formal-homework
+
+`formal-homework` is a small [Typst](https://typst.app) package that provides an easy way to start writing formal homework documents.
+
+
+## Usage
+
+First, create a new `.typ` file and import this package at the top:
+
+```typst
+#import "@preview/formal-homework:0.1.0": hw, q, a, br
+```
+
+The entirety of the homework portion of your document will be contained in `hw()[]`, including the title page. Call it and pass the following (optional) parameters:
+
+- `title-text` -> Text to be used as title of document
+
+- `number` -> Number of the homework, only used if `title_text` is omitted
+
+- `author` -> Your name
+
+- `class` -> Name of the class/course that the homework is for
+
+- `due-date` -> Date that the homework is due
+
+```typst
+#hw(
+  number: 5,
+  author: "George P. Burdell",
+  class: "CS 2050",
+  due-date: "January 1, 1970"
+)[
+    Your content goes here.
+]
+```
+
+![Title page example](images/title.png)
+
+New Computer Modern is the default font, aiming for semblance to vanilla LaTeX, which professors likely prefer. To revert it, insert `#set text(font: "libertinus serif")` into the body of `#hw()[]`.
+
+The content of the document is laid out with the remaining macros: `#q[]`, `#a[]`, and `#br()`
+
+`#q[]` -> Contains the question, automatically enumerated
+`#a[]` -> Contains the answer, bordered with a black box
+`#br()` -> Shortcut for `#pagebreak()`, may be used to keep questions on their own page
+
+Note that `#q[]`s may embed `#a[]`, which is often useful for multi-part questions. An exhaustive example is provided below.
+
+```typst
+
+#q[
+    Let $f: ZZ^+ times ZZ^+ -> ZZ^+$ such that $f(m,n) = 2^(m - 1) (2n - 1)$. Prove that $f$ is bijective.
+]
+
+#a[
+    *Lemma:* #h(0.5em) We proceed directly to show that the product of an even integer and an odd integer is an even integer.
+
+    Let $x$ be an odd integer. By definition of odd integer, $x = 2k + 1$ for some integer $k$. Let $y$ be an even integer. By definition of even integer, $y = 2j$ for some integer $j$. We show that the product of $x$ and $y$ is even.
+
+    $
+    x y &= (2k + 1)(2j) \
+    &= 4 k j + 2 j \
+    &= 2 (2 k j + j) \
+    $
+
+    Since integers are closed under multiplication and addition, the expression $2 k j + j$ is an integer. Let that expression be assigned the variable $z$. Since $x y = 2z$, the product of $x$ and $y$ is an even integer. Therefore, the product of an even integer and an odd integer is an even integer. #sym.qed
+
+    *Proof:* #h(0.5em) We proceed directly to show that the function $f: ZZ^+ times ZZ^+ -> ZZ^+$ such that $f(m, n) = 2^(m - 1) (2n - 1)$ is bijective.
+
+    Let $f: ZZ^+ times ZZ^+ -> ZZ^+$ such that $f(m, n) = 2^(m - 1) (2n - 1)$. To show that $f$ is bijective, we must show injectivity and surjectivity.
+
+    *Injectivity Proof:* #h(0.5em) To show that $f$ is injective, we show that if $f(a,b) = f(c,d)$ for arbitrary $(a,b), (c,d) in (ZZ^+ times ZZ^+)$, then $(a,b) = (c,d)$.
+
+    Assume that for arbitrary $(a,b), (c,d) in (ZZ^+ times ZZ^+)$, $f(a,b) = f(c,d)$.
+
+    $ 
+    f(a,b) &= 2^(a - 1) (2b - 1) \
+    f(c,d) &= 2^(c - 1) (2d - 1) \
+    $
+    $
+    &=> 2^(a - 1) (2b - 1) = 2^(c - 1) (2d - 1) \
+    &= 2^(a - c) (2b - 1) = 2d - 1 \
+    $
+
+    By Lemma, the product of an even integer and a positive integer is an even integer. Since $2b - 1$ and $2d - 1$ are odd integers, $2^(a - c)$ cannot be even. If $2^(a - c)$ was even, then $2d - 1$ would be even by Lemma, which is a contradiction. Thus, $2^(a - c)$ must be odd, and the only power of two that is odd is $2^0 = 1$. It follows that $2^(a - c) = 2^0$, so $a = c$. 
+
+    Since $a = c$, we plug $a$ back into $f(a,b) = f(c,d)$ for $c$.
+
+    $
+    2^(a - 1) (2b - 1) &= 2^(a - 1) (2d - 1) \
+    2b - 1 &= 2d - 1 \
+    2b &= 2d \
+    b &= d
+    $
+
+    Since $a = c$ and $b = d$, we have shown that if $f(a,b) = f(c,d)$, then $(a,c) = (c,d)$. Therefore, $f$ is injective.
+
+    *Surjectivity Proof:* #h(0.5em) To show that $f$ is surjective, we prove that for an arbitrary positive integer $z$, there exists an element $(x, y) in (ZZ^+ times ZZ^+)$ such that $z = f(x, y)$.
+
+    Let $z$ be an arbitrary positive integer. By the Fundamental Theorem of Arithmetic, $z$ may be uniquely written as $z = 2^p q$, where $p$ is a non-negative integer and $q$ is an odd positive integer. By the definition of odd positive integers, $q$ may be expressed as $q = 2k + 1$, where $k$ is a non-negative integer. Expressing $z$ in terms of $p$ and $k$ gives $z = 2^p (2k + 1)$. Define the variables $x = p + 1$ and $y = k + 1$. By the closure of integers under addition, $x$ and $y$ must be integers. Since $p$ and $q$ are positive, $x$ and $y$ must be positive. Therefore, $z$ may be expressed as a function of $x$ and $y$, and $(x,y)$ is an element of $ZZ^+ times ZZ^+$ by definition of Cartesian product.
+
+    Therefore, since we have shown that the function $f: ZZ^+ times ZZ^+ -> ZZ^+$ such that $f(m, n) = 2^(m - 1) (2n - 1)$ is injective and surjective, $f$ is bijective by definition of bijectivity. #sym.qed
+]
+
+#br()
+
+#q[
+    Determine whether the following sets are finite, countably infinite, or uncountably infinite. If finite, find it's cardinality. If countable, construct an injection from it to $NN $ (you do not have to prove that your function is an injection).
+
+    +  The set of all multiples of 6.
+
+      #a[Countably infinite. For $s = 6k$, where $k in ZZ$, let $f(s) = |s| + 1$.]
+
+    +  The set of all even prime numbers.
+
+      #a[Finite. The only even prime is 2; this set has a cardinality of 1.]
+
+    +  The set of finite binary strings. Note: a binary string consists of $0 $s and $1 $s, for example: $1 0 1 1 $ and $0 0 1 0 0 0 0 1 1 $ are binary strings.
+
+      #a[Countably infinite. Let $f$ be a function that takes a binary string as input and outputs the base 10 conversion of a 1 followed by the binary string. ]
+
+    +  The set of all infinite binary strings.
+
+      #a[Uncountably infinite.]
+
+    +  The set of all real numbers between $e $ and $pi $ (inclusive).
+
+      #a[Uncountably infinite.]
+
+    +  The set of all integers between $e $ and $pi $ (inclusive).
+
+      #a[Finite. This set has a cardinality of 1, with its only element being 3.]
+
+]
+```
+
+![Proof page 1 example](images/proof1.png) ![Proof page 2 example](images/proof2.png) ![Subquestions example](images/subquestions.png)
+

packages/preview/formal-homework/0.1.0/lib.typ

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+#let styles = ("1.", "(a)", "(i)")
+
+#let enum-offset = state("offset", 0)
+
+#let q-counter = counter("q")
+
+#let q(body) = [
+  #q-counter.step()
+
+  #context {
+    let num = q-counter.display("1.")
+    enum-offset.update(1)
+
+    block(width: 100%, above: 0.75em, below: 1em)[
+      #place(left)[#box(width: 0.5em, align(right, num))]
+      #pad(left: 1.3em, body)
+    ]
+
+    enum-offset.update(0)
+  }
+]
+
+#let a(body) = [
+  #block(
+    width: 100%,
+    // In the future, these should be optional arguments 
+    // fill: none,
+    // radius: 0pt,
+    inset: 10pt,
+    stroke: 1pt + black,
+    breakable: true,
+  )[#body]
+]
+
+#let br() = pagebreak()
+
+
+#let hw(
+  title-text: none,
+  number: 1, 
+  author: none, 
+  class: none, 
+  due-date: none, 
+  body,
+) = {
+
+  counter(page).update(0)
+
+  set page(
+    paper: "us-letter",
+    margin: 1in,
+    footer: context {
+      if counter(page).get().first() > 0 [
+        #align(center)[
+          #counter(page).display()
+        ]
+      ]
+    },
+  )
+
+  set text(
+    size: 12pt,
+    lang: "en",
+    font: "New Computer Modern"
+  )
+
+  set enum(
+    full: true,
+    numbering: (..nums) => context {
+      let offset = enum-offset.get()
+      let depth = nums.pos().len() + offset
+      let style = styles.at(depth - 1)
+      numbering(style, nums.pos().last())
+    }
+  )
+
+
+  align(horizon + center, block[
+    #if title-text != none [
+      #title[#title-text]
+    ] else [
+      #title[Homework #number Submission]
+    ]
+
+    #class
+
+    #author
+
+    #due-date
+  ])
+
+
+  pagebreak()
+
+  body
+}

packages/preview/formal-homework/0.1.0/typst.toml

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+[package]
+name = "formal-homework"
+version = "0.1.0"
+entrypoint = "lib.typ"
+
+authors = ["<@daqnal>"]
+license = "GPL-3.0-or-later"
+description = "Submit homeworks with simplicity and formality."
+
+repository = "https://github.com/daqnal/formal-homework"
+categories = ["report"]