Notation Reference

A rigorous notation reference for algebra, sets, functions, intervals, vectors, calculus, probability, and logic.

Notation becomes meaningful only when symbol, role, units, and conditions are connected. A central expression connects to four questions: what each symbol names, what operation it signals, which units apply, and which conditions limit its use.

Precise definition

Notation assigns compact symbols to defined mathematical objects and relations. A reference should record syntax, meaning, type, scope, and conditions. Examples: $f:A\to B$ is a function; $f^{-1}$ can mean inverse function only when appropriate; $A^{-1}$ is a matrix inverse; $x^{-1}=1/x$ for $x\ne0$.

Notation and mathematical language

Set notation includes $\in,\notin,\subseteq,\cup,\cap,\varnothing$. Interval $[a,b)$ includes $a$ and excludes $b$. Calculus uses $f'(x)$, $dy/dx$, $\int_a^bf(x)dx$, and limits. Probability uses $P(A\mid B)$ with $P(B)>0$, expectation $E[X]$, and variance $\operatorname{Var}(X)$.

Conceptual picture

Notation is typed grammar. A scalar, vector, set, function, and proposition support different operations. Parentheses and binding conventions determine scope; $-x^2$ means $-(x^2)$ while $(-x)^2=x^2$.

Fully worked example

Interpretation and application

Notation fluency supports textbooks, proofs, software, and communication across fields. A reference is most effective when every entry includes an example and a near-confusable nonexample.