TeXipedia

coloneq

Represents a colon followed by an equals sign, commonly used to denote definitions or assignments in mathematical notation.

Overview

Serves as a specialized relational operator that combines a colon and equals sign to indicate formal definitions, assignments, or specifications in mathematical writing.

  • Frequently used in formal mathematics and computer science to distinguish definitions from equality statements
  • Appears in set theory and abstract algebra when defining new terms or functions
  • Provides a more formal alternative to the standard equals sign when explicitly stating definitions
  • Often encountered in theoretical computer science and mathematical logic for formal specifications

Examples

Defining a sequence in number theory

an:−2n−1a_n \coloneq 2^n - 1 
a_n \coloneq 2^n - 1

Defining a function by cases

f(x):−{x2x≥0−x2x<0f(x) \coloneq \begin{cases} x^2 & x \geq 0 \\ -x^2 & x < 0 \end{cases} 
f(x) \coloneq \begin{cases} x^2 & x \geq 0 \\ -x^2 & x < 0 \end{cases}

Defining a set with specific properties

S:−{x∈R:x2>2}S \coloneq \{x \in \mathbb{R} : x^2 > 2\} 
S \coloneq \{x \in \mathbb{R} : x^2 > 2\}