A field guide to the symbols

Read the sentence inside the symbol

Notation saves space. It does not replace meaning. When a symbol looks unfamiliar, ask: “What kind of object is this? What operation is happening? What are the inputs?”

Explore it yourself

A symbol is a compressed sentence

Choose a symbol. Read its meaning before trying to memorize its shape.

Preparing interactive controls…

ℝ

Real numbers

The set of real numbers.

−2, 0, ½, √2, and π are real numbers.

Objects: where the numbers live

The following tables preserve every entry in the source notation table. The examples and reading advice are teaching additions.

Source symbol	Meaning	A small example
$\mathbb R$	The set of real numbers	$-2$ , $1/2$ , and $\sqrt2$ belong to it.
$\mathbb R^n$	The vector space of real n-tuples, endowed with the usual inner product	$(2,1,-3)\in\mathbb R^3$ ; the usual inner product is the sum of component-wise products.
$\mathbb R^{m\times n}$	The vector space of real m-by-n matrices	Two rows and three columns gives a $2\times3$ matrix.

The superscript in $\mathbb R^3$ counts coordinates. It does not mean “cube each number.” The $m\times n$ specifies rows first, columns second.

Operations: what to do with an object

Source symbol	Meaning	A small example
$\delta_{ij}$	Kronecker delta: 1 when $i=j$ , 0 otherwise	$\delta_{22}=1$ and $\delta_{23}=0$ .
$\nabla f(\mathbf x)$	Gradient of $f$ at $\mathbf x$	For $f(x_1,x_2)=x_1^2+3x_2$ , the gradient is $(2x_1,3)^\top$ .
$\nabla^2 f(\mathbf x)$	Hessian of $f$ at $\mathbf x$	For the same function, the second-derivative matrix is $\begin{bmatrix}2&0\\0&0\end{bmatrix}$ .
$\mathbf A^\top$	Transpose of matrix $\mathbf A$	Its $(i,j)$ entry becomes the $(j,i)$ entry.

The 2 in $\nabla^2$ signals second derivatives; it does not mean squaring each entry of the gradient. Kronecker delta is an equality indicator, not a derivative or an arbitrarily small change.

Probability: describe uncertainty

Source symbol	Meaning	A small example or reading cue
$\Omega$	Sample space	For a coin flip, $\{H,T\}$ .
$P(A)$	Probability of event A	$P(\{H\})=1/2$ for a fair coin.
$p(X)$	Distribution of random variable X	The whole probability description of X.
$p(x)$	Probability mass/density function evaluated at x	The value of the relevant function at a particular input.
$A^c$	Complement of event A	All outcomes in $\Omega$ outside A.
$A\mathbin{\dot\cup}B$	Union with the extra requirement $A\cap B=\varnothing$	For disjoint events, no outcome belongs to both.
$E[X]$	Expected value of X	A probability-weighted average.
$\operatorname{Var}(X)$	Variance of X	Average squared distance from its mean.
$\operatorname{Cov}(X,Y)$	Covariance of X and Y	Average product of their deviations from their means.

The interactive picker uses $\sqcup$ as a readable alternate mark for disjoint union; the source uses a dotted union. The condition “no overlap” is essential with either notation.

One capital letter can change the question

Let $X$ be the outcome of a fair die, and let $x=3$ be one particular value. The notation $p(X)$ refers to the distribution over all six outcomes; $p(3)=1/6$ evaluates its mass function at one outcome. $P(X\in\{2,4,6\})=1/2$ asks about an event containing three outcomes.

For a continuous variable with a density, density is not point probability. A density can exceed 1. Probabilities come from integrating over intervals. Module 7 will explain this carefully; the symbol table introduces the distinction now.

Make a prediction

Which expression refers to the entire distribution in the source’s convention?

Context beats memorization

Capital Roman letters can name matrices or random variables. Read the surrounding definition. The notation $A$ in “ $P(A)$ ” names an event; bold $\mathbf A$ usually names a matrix. The next lesson makes these conventions explicit.