ECS132 LaTeX Math Reference

LaTeX can be daunting to get started with, and the documentation dense. I put this notepad together to demo a subset of LaTeX relevant to the course; and because it's on HackMD, it also demos some Markdown at the same time. I hope it helps you out!

Entering Math Mode

To enter math mode, surround your LaTeX code with $ for inline and $$ for long form. For example, $P$ gives you \(P\), while $$P$$ will put it on its own line.

You can also use the LaTeX commands directly:

\begin{equation}
P(X)
\end{equation}

renders as:

\begin{equation*} P(X) \end{equation*}

Most Markdown-based editors – like Jupyter – will recognize both forms.

Some Simple Equations

Many basic equations don't require any special syntax. Upper-case letters will be converted to their "mathy" formats, and operations and symbols will be properly spaced.

• $P(A)-1$: \(P(A)-1\)
• $P(A)+P(B)$: \(P(A)+P(B)\)
• $P(A)P(B)$: \(P(A)P(B)\)

Some symbols need to be escaped with \ because they have special meaning to LaTeX. For example, curly braces: $S = \{1,2,3,4,5,6\}$: \(S = \{1,2,3,4,5,6\}\).

Some operations have special commands. For example, the "x" multiplication symbol is invoked with \times: $A \times B$ gives \(A \times B\).

Subscripts and Superscripts

Subscripts are defined with _ and superscripts with ^. For example:

• $A_i$: \(A_i\)
• $A^i$: \(A^i\)

If the sub or superscript has multiple symbols, it needs to be fenced with {}.

• $(1-p)^{k-1}$: \((1-p)^{k-1}\)
• $A_{i-1}$: \(A_{i-1}\)

Fractions

Long-form fractions use the \frac{}{} command: the numerator goes in between the first braces and denominator in the second. For example, $$\frac{n!}{(n-k)!k!}$$ gives you: \[\frac{n!}{(n-k)!k!}\]

For fractions with braces, we can use the \left( and \right) commands. Note that both are necessary. For example, $$\left( \frac{n-1}{n} \right)^(k-1)$$ gives: \[\left( \frac{n-1}{n} \right)^{k-1}\]

Also note that I threw an exponent on there by tacking on ^{k-1} – LaTeX is smart enough to apply it to the entire braced expression.

Counting

Counting notation works similarly, but the command is either \choose or \binom{}{}. For example, $$\binom{n}{k}$$ gives: \[\binom{n}{k}\]

and we get the same with $$n \choose k$$: \[n \choose k\]

Note that, with \choose, we need to wrap it in braces if there are other terms in order to avoid ambiguity. So, more precisely, we should do $${n \choose k}$$.

Set Notation

We use intersection and union a lot in this class! These two symbols, like many other math operators, have their own commands:

• Union is \cup, like $A \cup B$: \(A \cup B\)
• Intersection is \cap, like $A \cap B$: \(A \cap B\)

We also want to be able to donate membership.

• "Is an element of" is \in, like $a_i \in A$: \(a_i \in A\)
• "Is not an element of" is \notin, like $x \notin A$: \(x \notin A\)

And of course, sub- and supersets:

• "Is a subset of" is \subseteq, like $V \subseteq W$: \(V \subseteq W\)
• "Is a proper subset of" is \subset, like $V \subset W$: \(V \subset W\)

Comparisons

The comparison operators follow a somewhat standard format, with equals, less than, and greater than using their raw symbols and the combined operators having their own commands:

• equals uses the raw symbol =: $A = B$ gives \(A = B\)
• less than uses the raw symbol <: $A < B$ gives \(A < B\)
• greater than uses the raw symbol >: $A > B$ gives \(A > B\)
• less than or equal to uses \leq or \leq: $A \leq B$ gives \(A \leq B\)
• greater than or equal to is similar, \geq or ge: $A \geq B$ gives \(A \geq B\)
• not equal to is, you guess it, \neq: $A \neq B$ gives \(A \neq B\)

The On-Line Encyplopedia of Integer Sequences has a nice chart of many additional unary and binary operators for your reference.

Additional Notation

The complement symbol has its own command, \complement. We can use it along with the superscript command like so:

• $P(A^\complement) = 1 - P(A)$: \(P(A^\complement) = 1 - P(A)\)

Conditional probability uses a raw pipe | symbol. So $$P(A|B) = \frac{P(A)P(B|A)}{P(B)}$$ (Baye's Rule!) gives: \[P(A|B) = \frac{P(A)P(B|A)}{P(B)}\]

Compound Operators

Summation combines the sub and superscript syntax with its own command \sum. Intuitively, below the sum uses subscript, and above uses superscript. For example, $$\sum_{i=0}^{10} n_i$$ gives: \[\sum_{i=0}^{10} n_i\]

Products work similarly. For example, $$\prod_{i=0}^{k} n-i$$ gives: \[\prod_{i=0}^{k} n-i\]

Integration uses the \int command; $$\int_{a}^{\infty} f_X (x) dx$$ gives: \[\int_{a}^{\infty} f_X (x) dx\]

Note how I slipped in that \infty to get \(\infty\).

Going Further

Note that all of these commands can be combined together! Just make sure to keep track of how many braces you've used – just like in any other programming language (and yes, depressingly, LaTeX is Turing-complete), things will go haywire if you mismatch braces. Some examples:

• \frac and \binom: $$\frac{1}{\binom{52}{13}}$$ gives \[\frac{1}{\binom{52}{13}}\]

Equation Arrays

We can create (semi)-aligned arrays of equations with the \eqnarray command:

\begin{eqnarray}
  a & = & b + c \\
  x & = & y - z
\end{eqnarray}

\[ \begin{eqnarray} a & = & b + c \\ x & = & y - z \end{eqnarray} \]

The & acts as a column-separator within each row, setting the alignment; the \\ denotes the end of a row.

Explicit Bracket Sizes

Sometimes you want finer control over bracket sizes. This can be achieved with the "big" family of commands: $$\big\{ \big\} \Big\{ \Big\} \bigg\{ \bigg\} \Bigg\{ \Bigg\}$$ renders as: \[\big\{ \big\} \Big\{ \Big\} \bigg\{ \bigg\} \Bigg\{ \Bigg\}\]

Stick whichever sort of bracket after the \big as you want; for example, you can make a really big pipe with $\Bigg|$: \(\Bigg|\)

Additional Resources

• Detexify: Martin pointed out this great resource that allows you to draw a symbol which will be run through a classifier to guess the appropriate command: https://detexify.kirelabs.org/classify.html. There is also an Android app on the Play Store.