Last modified: 2026-09-28 23:45:45 (PDT)
Mathematical notation is not standardized. This section states the conventions these notes use, and the alternatives you may meet in other sources.
| symbol | meaning | LaTeX |
|---|---|---|
| \(\neg\) | not | \neg |
| \(\forall\) | all | \forall |
| \(\exists\) | some | \exists |
| \(\cup\) | union, “or” | \cup |
| \(\cap\) | intersection, “and” | \cap |
| \(\mid\) | given, conditional on | \mid, | |
| \(\sum\) | sum | \sum |
| \(\prod\) | product | \prod |
| \(\mu\) | mean | \mu |
| \(\operatorname{E}\) | expectation | \mathbb{E} |
| \(x^{\top}\) | transpose of \(x\) | x^{\top} |
| \('\) | transpose or derivative1 | ' |
| \(\perp\!\!\!\perp\) | independent | \perp\!\!\!\perp |
| \(\therefore\) | therefore, thus | \therefore |
| \(\eta\) | linear component of a GLM | \eta |
| \(\mathopen{}\left\lfloor x\right\rfloor\mathclose{}\) | floor of \(x\): largest integer less than or equal to \(x\) | \lfloor x \rfloor |
| \(\mathopen{}\left\lceil x\right\rceil\mathclose{}\) | ceiling of \(x\): smallest integer greater than or equal to \(x\) | \lceil x \rceil |
| \(\mathbb{1}_{A}(x)\), \(\mathbb{1}\mathopen{}\left(P\right)\mathclose{}\) | indicator function (Section 5): \(1\) if condition holds, \(0\) otherwise | \indic{A}(x), \indicp{P} |
The third column of Table 1 gives the LaTeX command for each symbol. Quarto and R Markdown documents write math in LaTeX syntax: put inline math between single dollar signs, as in $x^2$, and displayed equations between double dollar signs, as in $$x^2$$. The equations section of Quarto’s Markdown guide shows the details. These notes also use shorthand macros, such as \floor{x} for \(\mathopen{}\left\lfloor x\right\rfloor\mathclose{}\), defined in the latex-macros submodule’s macros.qmd.
Exercise 1 (Which numbers are natural?) List the elements of the set \(\mathopen{}\left\{n \in \mathbb{N} : n < 3\right\}\mathclose{}\). Is your answer the same in every textbook?
Solution 1. The answer depends on whether the source counts \(0\) as a natural number:
Both conventions are in common use, so the answer is not the same in every textbook.
Definition 1 (Natural numbers (our convention)) In these notes, the natural numbers are the positive integers:
\[\mathbb{N} \stackrel{\text{def}}{=}\mathopen{}\left\{1, 2, 3, \ldots\right\}\mathclose{}\]
Definition 2 (Non-negative integers) The non-negative integers are the natural numbers (Definition 1) together with \(0\):
\[\mathbb{N}_0 \stackrel{\text{def}}{=}\mathopen{}\left\{0, 1, 2, 3, \ldots\right\}\mathclose{} = \mathbb{N} \cup \mathopen{}\left\{0\right\}\mathclose{}\]
Example 1 (Natural numbers in a data analysis)
Other conventions for the natural numbers
Other sources may define \(\mathbb{N}\) differently, so check each source’s definition before reading its formulas:
When a formula’s meaning depends on whether \(0\) is included, we write the set out explicitly, for example \(\mathopen{}\left\{0, 1, 2, \ldots\right\}\mathclose{}\) or \(\mathopen{}\left\{1, \ldots, n\right\}\mathclose{}\).
Source: Wikipedia, “Natural number”, “Terminology and notation” and “Zero as natural number”, which cites ISO 80000-2:2019 and the textbooks using each notation.
The percent sign “%” is just a shorthand for “\(/100\)”. The word “percent” comes from the Latin “per centum”; “centum” is Latin for 100, so “percent” means “per hundred” (cf. https://en.wikipedia.org/wiki/Percentage)
So, contrary to what you may have learned previously, \(10\% = 0.1\) is a true and correct equality, just as \(10 \text{kg} = 10,000 \text{g}\) is true and correct.
Proof. \[ \begin{aligned} 10\% &= 10 / 100 \\ &= \frac{10}{100} \\ &= 0.1 \end{aligned} \]
You are welcome to switch between decimal and percent notation freely; just make sure you execute it correctly.
We can use any of:
\therefore in LaTeX),\Rightarrow),\models)to denote logical entailments (deductive consequences).
Let’s save \(\rightarrow\) (\rightarrow) for convergence results.
An indicator function is a mathematical function that signals whether an element belongs to a specified set, or whether a given logical condition is satisfied. In statistics and epidemiology, indicator functions are ubiquitous: they represent binary variables, censor and event indicators in survival analysis, membership in subpopulations, and domain restrictions in integrals and sums.
Despite their conceptual simplicity, notation for indicator functions varies substantially across textbooks, research papers, and subfields. This section summarizes the principal notational conventions.
Definition 3 (Indicator function) For any subset \(A \subseteq \Omega\) of a universal set \(\Omega\), the indicator function of \(A\) is the function \(\mathbb{1}_{A} : \Omega \to \{0, 1\}\) defined by:
\[ \mathbb{1}_{A}(x) \stackrel{\text{def}}{=}\begin{cases} 1, & x \in A \\ 0, & x \notin A \end{cases} \]
More generally, for any logical proposition or predicate \(P\), the indicator of \(P\) takes the value \(1\) when \(P\) is true and \(0\) when \(P\) is false:
\[ \mathbb{1}\mathopen{}\left(P\right)\mathclose{} \stackrel{\text{def}}{=}\begin{cases} 1, & \text{if } P \text{ is true} \\ 0, & \text{if } P \text{ is false} \end{cases} \]
Example 2 (Evaluating set and predicate indicators) Consider the real line \(\Omega = \mathbb{R}\), the set of nonnegative numbers \(A = [0, \infty)\), and a continuous random variable \(Y\).
The vast majority of indicator notations belong to one of two families: set notation or predicate notation.
In set notation, the indicator is tied to a set \(A\), which appears as a subscript:
When the function is viewed as a mathematical object in its own right (for instance, as an element of an \(L^p\) function space), authors often omit the argument \(x\), writing simply \(\mathbf{1}_A\), \(\mathbb{1}_A\), or \(I_A\).
In predicate notation, the indicator takes a logical condition, relation, or proposition \(P\) directly as its argument or subscript:
Predicate notation is especially common in applied statistics and survival analysis, where indicators frequently depend on inequalities involving random variables, such as \(\mathbb{1}\mathopen{}\left(T_i \le t\right)\mathclose{}\) (an event occurring before time \(t\)) or \(\mathbb{1}\mathopen{}\left(Y_i = 1\right)\mathclose{}\) (a binary outcome).
The two paradigms are connected by evaluating the predicate indicator at the membership statement \(x \in A\):
\[ \mathbf{1}_A(x) = \mathbb{I}(x \in A) \]
Set notation is more natural when the underlying set \(A\) has a standard name (such as the support of a distribution or a geometric region). Predicate notation is more natural when the condition involves compound inequalities, such as \(\mathbb{1}\mathopen{}\left(0 \le t \le u\right)\mathclose{}\).
In 1962, Kenneth Iverson introduced a compact notation in the programming language APL, later popularized in mathematics and computer science by Donald Knuth: the Iverson bracket.
Definition 4 (Iverson bracket) For any logical proposition \(P\), the Iverson bracket of \(P\) is
\[ [P] \stackrel{\text{def}}{=}\begin{cases} 1, & \text{if } P \text{ is true} \\ 0, & \text{if } P \text{ is false} \end{cases} \]
Example 3 (Evaluating Iverson brackets)
Definition 5 (Kronecker delta) For integers \(i\) and \(j\), the Kronecker delta is
\[\delta_{ij} \stackrel{\text{def}}{=}[i = j]\]
that is, \(\delta_{ij} = 1\) when \(i = j\) and \(\delta_{ij} = 0\) when \(i \neq j\) (Definition 4).
Example 4 (Evaluating the Kronecker delta)
The primary advantage of the Iverson bracket is algebraic conciseness: it converts domain restrictions in sums and integrals into unrestricted operations. For example:
\[ \sum_{x \in A} f(x) = \sum_{x} f(x) [x \in A] \]
With \(A = \mathopen{}\left\{2, 4\right\}\mathclose{}\), \(f(x) = x\), and \(x\) running over \(\mathopen{}\left\{1, 2, 3, 4, 5\right\}\mathclose{}\), both sides equal \(2 + 4 = 6\).
However, in statistics and epidemiology, square brackets are already heavily overloaded: they denote closed intervals \([a, b]\), conditional expectations \(\operatorname{E}[Y \mid X]\), and matrix delimiters. To prevent visual confusion with expectation brackets or intervals, statistical literature predominantly uses \(\mathbb{1}\) or \(I\) rather than the bare Iverson bracket.
Table 2 compares the major notations encountered across the literature.
| Notation style | Typical syntax | Primary fields | Notes and potential ambiguities |
|---|---|---|---|
| Blackboard bold 1 | \(\mathbb{1}_A(x)\), \(\mathbb{1}(P)\) | Modern probability, mathematical statistics | Unambiguous; distinct from matrices and scalars; standard in this book. |
| Bold numeral 1 | \(\mathbf{1}_A(x)\), \(\mathbf{1}(P)\) | Probability theory, measure theory | Can be confused with a vector of ones \(\mathbf{1} = (1, \dots, 1)^{\top}\). |
| Blackboard bold I | \(\mathbb{I}(x \in A)\), \(\mathbb{I}(P)\) | Econometrics, machine learning, statistics | Clear predicate notation; avoids confusion with numerals. |
| Letter \(I\) | \(I_A(x)\), \(I(P)\) | Classical statistics, epidemiology | Can be confused with the identity matrix \(I\) or Fisher information \(\mathcal{I}\). |
| Iverson bracket | \([P]\), \([x \in A]\) | Computer science, discrete mathematics | Very compact, but square brackets collide with intervals and expectation brackets. |
| Greek letter \(\chi\) | \(\chi_A(x)\) | Real analysis, measure theory | Often termed “characteristic function”; collides with the Fourier transform in probability. |
In these notes, we standardize on blackboard bold \(\mathbb{1}\) via the macros defined in latex-macros/macros.qmd:
\indic{A} produces \(\mathbb{1}_{A}\) (set subscript)\indicp{P} produces \(\mathbb{1}\mathopen{}\left(P\right)\mathclose{}\) (predicate in parentheses)\indiccb{P} produces \(\mathbb{1}\mathopen{}\left\{P\right\}\mathclose{}\) (predicate in curly braces)\1{P} produces \(\text{1}_{P}\) (text numeral with subscript, used in legacy formulas)Blackboard bold \(\mathbb{1}\) is preferred because it avoids all common collisions: it is visually distinct from the scalar \(1\), the identity matrix \(I\), and the information matrices (\(I\), \(\mathcal{I}\)).
Indicator functions translate logical operations on events into ordinary arithmetic on real numbers:
For subsets \(A\) and \(B\) of \(\Omega\), with complement \(A^c \stackrel{\text{def}}{=}\Omega \setminus A\), and for every \(x \in \Omega\):
Intersection (“and”): \(\mathbb{1}_{A \cap B}(x) = \mathbb{1}_{A}(x) \cdot \mathbb{1}_{B}(x)\)
Union (“or”): \(\mathbb{1}_{A \cup B}(x) = \mathbb{1}_{A}(x) + \mathbb{1}_{B}(x) - \mathbb{1}_{A}(x) \cdot \mathbb{1}_{B}(x)\)
Complement (“not”): \(\mathbb{1}_{A^c}(x) = 1 - \mathbb{1}_{A}(x)\)
Idempotence: \((\mathbb{1}_{A}(x))^2 = \mathbb{1}_{A}(x)\)
Expectation gives probability: For any event \(A\), the expectation of its indicator is the probability of the event:
\[ \operatorname{E}[\mathbb{1}_{A}] = 0 \cdot \Pr(A^c) + 1 \cdot \Pr(A) = \Pr(A) \]
This fundamental identity connects probability theory directly to linear expectation. It provides the mathematical foundation for empirical proportions, survival curve estimators, and regression models for binary outcomes.
In grad school, we are asked to learn from increasingly disorganized materials and lectures. Not coincidentally, as the amount of organization decreases, the amount of complexity increases, the amount of difficulty increases, the number of reliable references decreases, and the amount of inconsistency in notation and content increases (both between multiple references and within single references!). In other words, as you approach the cutting-edge of most fields, you start to run into content that hasn’t been fully thought through or standardized. This lack of clarity is unfortunate and undesirable, but it is understandable and inevitable.
It’s worth noting that calculus was formalized in the 1600s, elementary algebra was formalized around 820, and arithmetic even earlier. And calculus still has several competing notation systems. In contrast, the field of statistics only emerged in the late 1800s and early 1900s, so it’s not surprising that the notation and terminology is still developing. Generalized linear models were only formalized in 1972 (Nelder and Wedderburn 1972), which is very recent in terms of the pace of scientific development.