Overleaf and LaTex

Overleaf is a web application for collaboratively editing documents typeset using LaTeX.

LaTeX is a powerful and extensible typesetting language that allows you to produce beautiful, complex documents from structured text input. The philosophy of LaTex is for you to write without worrying about how things will look.

LaTeX was developed by Leslie Lamport (who won the 2013 Turing Award for work on distributed computing) as a set of macros built on TeX to make it easier to produce documents. TeX was developed by Donald Knuth (who won the 1974 Turing Award and was among the first people to formally analyze algorithms), because he was unsatisfied with the typesetting tools available for writing The Art of Computer Programming (which he started writing in 1962, but is still unfinished).

Basics

“source” and “rendering”

We will refer to LaTeX that we write as “source”, and we will refer to the result of compiling some LaTeX code as its “rendering”. In other words, this LaTeX source:

$P :=$ Latex is great

Is rendered as:

Macros

A macro in LaTeX is something that begins with \. This includes math commands like \times or non-math commands like \textbf. A macro can take zero or more parameters. Macros are analogous to functions in programming languages (but different from mathematical functions as we have discussed). An example of a macro with one input is \textbf{Bold} which will render as bold text.

The inputs to a macro follow the macro and each input is surrounded by curly braces ({}). If you don’t use curly braces around inputs, the command will only apply to the first character after the command. Curly braces act as a grouping method to ensure commands apply to a sequence of characters.

Comments

Comments in LaTeX start with the % character. You can comment out longer regions using a conditional (started with \iffalse and ended with if backwards \fi):

% $P :=$ Latex is great

\iffalse
This is a lot of junk that will not show up in the rendering!
\fi

Seperating Lines

One of the philosophies behind the design of LaTeX is that the structure of a document is separated from its presentation. So, most whitespace (including newlines) in the source do not influence the rendering.

To start a new paragraph in LaTeX use two new lines (i.e., a blank line):

Paragraph 1.
Second line in paragraph 1.

Paragraph 2. Second sentence in paragraph 2.
Third sentence in paragraph 2.

You can also use the \\ command to start a new line without starting a new paragraph:

Sentence 1, paragraph 1.\\
Sentence 2, paragraph 1.\\Sentence, paragraph 1.

New paragraph.

Given that we know
$a = 2b+c$
and that
$c\in{0,1}$,
then we can conclude that the statement is true.

Given that we know $a = 2b+c$ and that $c\in\{0,1\}$, then we can conclude that the statement is true.

Environments

An environment in LaTeX is a block of code between a \begin and an \end command pair. An environment has its own formatting rules. For example, in an enumerate environment, the \item command will create a new item in a numbered list. In an ‘itemize’ environment, the \item command creates a new item but in a bulleted (or some other symbol) list.

In an equation environment, code is automatically interpreted in math mode.

\begin{itemize}
   \item Text in this environment is formatted as a bulleted list.
   \item For cosemetic reasons, a list should always have at least two items, unlike a set which can be empty.
\end{itemize}
\begin{equation*}
f(x) = 2x+5
\end{equation*}

Because the math environment is so common, there are several different ways to denote it. The most compact is to use $ — text between two $ will be rendered as math.

Numbered lists

As we saw in class, a good way to write proofs is to separate your steps into a numbered list. Rather than numbering your items and numbering them manually, LaTeX comes with a neat tool for doing this dirty work for you.

Writing numbered lists

When writing a proof that uses multiple steps, you can use the enumerate environment in LaTeX.

Text before the enumerated list.
\begin{enumerate}
   \item The item command signals starting a new step. 
   Just like other parts of latex, an end-line character doesn't create a 
   new paragraph.
   
   You can write multiple paragraphs in a single item.
   \item Second step.
\end{enumerate}
Text after the enumerated list.

Nested lists (sublists)

Lists can be nested. Simply start a new enumerate environment at the point where you want the sublist to be created.

\begin{enumerate}
   \item first step
   \item second step. Includes the following substeps: 
   \begin{enumerate}
      \item substep 1.
      \item substep 2.
   \end{enumerate}
   still in the second step.
   \item substep 3.

\end{enumerate}

Referencing items in lists

If you want to reference an item in a list, you must give the item a label. This is done using the \label{} command:

   \item this step will be labeled "step-one"\label{step-one}

Now, you can refer to this item anywhere in your text using the \ref{} command by passing that item’s label.

\begin{enumerate}
   \item first step.\label{step-one}   
   \item second step. Includes the following 
   substeps: 
   \begin{enumerate}
      \item substep 1.\label{proof1:s1:ss1}
   \end{enumerate}

\end{enumerate}
References to items outside the list work, e.g. 
a reference to step \ref{step-one} or step 
\ref{proof1:s1:ss1}.

Math in latex

Math in latex is one of its strongest features. There are two modes for math in LaTeX: inline math mode, or display math mode. Inline math is when a mathematical equation is within a text paragraph, display math is when an equation is on its own line.

Inline math mode

You can add a mathematical equation to a paragraph in LaTeX by surrounding the equation with $ symbols.

This equation $x^2+5 = y$ is rendered 
in the middle of the paragraph.

Display math mode

If you want your equation to be prominent and take up the whole line, you can use display math mode. You can trigger display math using many environments. The simplest is the equation* environment.

The following equation will take up its own line
\begin{equation*}
   f(x) = x^{x}
\end{equation*}

\begin{enumerate}
   \item definitions.
   \item some case\begin{enumerate}
         \item a display math equation
         \begin{equation*}
            f(x|y) = xy-2
         \end{equation*}
      \end{enumerate}
\end{enumerate}

Splitting equations over multiple lines

Sometimes you want to show multiple steps of a derivation, or your equation is too long to fit in a single line. In such cases, you can use the align* enviornment. Inside this environment, you can separate lines using \\, which you add to every line except the last one. By default, consequent lines will be right-aligned.

\begin{align*}
   y = 3 + 2 - 3 +4 - 2\\
   + 4 +11 \\
   = 19
\end{align*}

Splitting equations over lines and align them

You can align the lines of a multi-line equation around certain points in a line. This is especially useful when simplifying an equation or moving things around an inequality. This is done using the align* environment. Inside the environment, split lines using \\ and specify where the symbols should align using the & symbol. You can think of & as a separator of columns in a table, where each & character within a line represents the beginning of a new column.

\begin{align*}
   y &= 3 + 2 - 3 
   +4 - 2\\ % Note that lines 1 and 2 are rendered 
   %as a single line
   & + 4 +11 \\
   & = 19
\end{align*}
\begin{align*}
   3+y &\geq x\\
   3 & \geq x-y
\end{align*}
\begin{align*}
   2x + 3 &= 11 \\
   2x &= 8 \\
   x &= 4
\end{align*}

Numbering and referencing equations

Sometimes, it is useful to number equations and reference them in your proof. This works for display math equations. To make your equations numbered, simply remove the * from the environment name.

\begin{equation}
x= y
\end{equation}
\begin{align}
x = 3y + z \\
+ 5v
\end{align}

You can give numbered equations a label and then reference them in your text. This is done through the \label{} and \ref{} commands. To reference an equation, add a \label{} command to the end of the equation (or the line for a multi-line equation) with your desired label, and then use \ref{} to refer to that equation.

Note: for multi-line equations using align, you can add \nonumber to lines in the equation that you don’t want to number, e.g., those that are a part of longer equations.

\begin{enumerate}
   \item definitions.
   \item some case\begin{enumerate}
         \item a display math equation
         \begin{equation}
            f(x|y) = xy-2\label{proof2:eq}
         \end{equation}
         \item two lines with different numbers 
         \begin{align}
            x &= 3y+5z+\sum_1^{10} v \nonumber\\
            & +(2\times 3)y \label{proof2:eq:start}\\
            x &= 1.5\times 2 \times y \label{proof2:eq:interm}
         \end{align}
      \end{enumerate}
   \item From equations \ref{proof2:eq} and \ref{proof2:eq:interm} we conclude ...
\end{enumerate}

Common math commands and operations

Writing math in inline or display math modes works very similarly. This section shows some of the operations you will need to use.

Common operators

Here is a list of commonly used operators in proofs

Here are some examples:

\begin{equation*}
\{x \in \mathbb{R} \mid x > 0\}    % set notation
\end{equation*}
\begin{equation*}
\forall x \in \mathbb{N} \exists y  % quantifiers
\end{equation*}
\begin{equation*}
A \subseteq B \implies A \cap C \subseteq B \cap C  % set logic
\end{equation*}

$$ {x \in \mathbb{R} \mid x > 0} % set notation $$ $$ \forall x \in \mathbb{N} \exists y % quantifiers $$ $$ A \subseteq B \implies A \cap C \subseteq B \cap C % set logic $$

Subscript and Superscript

Subscripts and superscripts (exponents) are done using the _ and ^ operators. Each operator makes the group that follows the operator a subscript or a superscript. Remember to always place the expression you want to make the subscript or superscript in curly braces.

$$
x_{2}, x_{2+1}, x^{2}, x^{2+1}
$$

$$ x_{2}, x_{2+1}, x^{2}, x^{2+1} $$

Fractions

Fractions can be written using the \frac{}{} command which takes two inputs, the numerator and the denominator, respectively.

$\frac{p}{q}$

$$ \frac{p}{q} $$

Nested fractions are possible, but pay attention to the braces.

\begin{equation*}
\frac{1}{\frac{2}{3}}
\end{equation*}

$$ \frac{1}{\frac{2}{3}} $$

Brackets

You can add brackets to your equations directly, except for curly braces, as those require an escape character.

\begin{equation*}
(a+1)
\end{equation*}
\begin{equation*}
[0,1]
\end{equation*}
\begin{equation*}
\{n\in\{1,2,3\} \mid n > 1\}
\end{equation*}

$$ (a+1) $$ $$ [0,1] $$ $$ \{n\in{1,2,3} \mid n > 1\} $$

Bracket resizing

Brackets are generally added using a single size regardless of their contents.

\begin{equation*}
(\frac{\frac{\frac{a}{b}}{c}}{d})
\end{equation*}

$$ (\frac{\frac{\frac{a}{b}}{c}}{d}) $$

To instruct LaTeX to resize brackets, use \left and \right when adding brackets.

\begin{equation*}
\left(\frac{\frac{\frac{a}{b}}{c}}{d}\right)
\end{equation*}
\begin{equation*}
\left(\sum_{i=1}^{10}\frac{i}{i/2}\right)
\end{equation*}

$$ \left(\frac{\frac{\frac{a}{b}}{c}}{d}\right) $$ $$ \left(\sum_{i=1}^{10}\frac{i}{i/2}\right) $$

Spaces

LaTeX takes care of adding spaces between characters in equations; you can’t add spaces in the rendering of an equation by adding extra spaces to the code

\begin{equation*}
a    =    b
\end{equation*}
\begin{equation*}
a=b 
\end{equation*}
\begin{equation*}
x y 
\end{equation*}

$$ a = b $$ $$ a=b $$ $$ x y $$

If you wish to manually add spaces, you need to use special commands.

\begin{equation*}
x \; y\\        % medium space
\end{equation*}
\begin{equation*}
x \: y\\        % thin space
\end{equation*}
\begin{equation*}
x \! y\\        % negative thin space
\end{equation*}

Example proofs

Here are fully written proofs as examples.

Example 1

Prove that for sets $A$, $B$, and $C$, if $A \subseteq B$ then 
$(A \cap C) \subseteq (B \cap C)$

\begin{enumerate}
    \item Let $A \subseteq B$ \label{step:subset}
    \item Take any $x \in (A \cap C)$ \label{step:element}
    \item From step \ref{step:element}, we know:
    \begin{enumerate}
        \item $x \in A$ \label{step:inA}
        \item $x \in C$ \label{step:inC}
    \end{enumerate}
    \item From steps \ref{step:subset} and \ref{step:inA}, we can conclude:
    \begin{equation}
        x \in B \label{eq:inB}
    \end{equation}
    \item From equation \ref{eq:inB} and step \ref{step:inC}, we have:
        \[ x \in B \land x \in C \implies x \in (B \cap C) \]
    \item Therefore, $(A \cap C) \subseteq (B \cap C)$
\end{enumerate}

Images

It is easy to incorporate images that you create with some other tool (e.g., taking a picture of a hand drawn image or a drawing using digitial ink in PowerPoint) in a LaTeX document. The best way to do this is to just make a PDF file of the image (although this works with most image formats such as PNG or JPG files) and upload the image to your overleaf repository. Then, you can include it in the LaTeX file using \includegraphics with the name of the image file as a parameter:

\begin{center}
\includegraphics[width=0.8\linewidth]{setdrawing.pdf}
\end{center}

The optional [width=0.8\linewidth] scales the image to $\frac{8}{10}$ of the width of the column.

Tables

Create tables using the tabular environment. The environment takes an input which will contain a letter for each column. Letters can be l, r, or c which make column left-, right-, or center-aligned.

The following declaration creates a table with three columns, the first two left-aligned and the last one center-aligned.

\begin{tabular}{llc}
%...
\end{tabular}

Rows inside a table are separated with \\ (except the last row), and columns within a row are separated with &. Note that you can use inline math inside a table.

\begin{tabular}{llc}
Name & ID & Salary \\
John & 
1234 & 
$9000.00$ \\ 
Doe &
5678 & $1000.00$
\end{tabular}

Adding visual separators

You can add vertical separators using | characters defined inside the tabular input braces.

\begin{tabular}{||l|l|c}
c1 & c2 & c3
\end{tabular}

You can add horizontal separators using the \hline command.

\begin{tabular}{||l|l|c}
\hline
Heading 1 & Heading 2 & Heading 3\\
\hline\hline
c1 & c2 & c3
\end{tabular}

Note that you shouldn’t ever actually make tables this ugly! A good table has as few horizontal and vertical lines as possible.