TeXipedia

Coloneqq

Represents a definition or assignment with a colon-equals symbol, commonly used in mathematical writing to introduce formal definitions.

Overview

Provides a more formal and visually distinct alternative to the standard equals sign when defining new mathematical concepts, variables, or functions.

  • Popular in theoretical mathematics and computer science for expressing definitions rigorously
  • Often preferred over := or = in academic papers and textbooks for its clarity
  • Particularly common in formal proofs, set theory, and abstract algebra where precise notation is essential
  • Helps distinguish definitional equations from regular equality statements

Examples

Defining a mathematical sequence explicitly.

an∷=2n+1a_n \Coloneqq 2^n + 1 
a_n \Coloneqq 2^n + 1

Defining a set using set-builder notation.

S∷={x∈R:x>0}S \Coloneqq \{x \in \mathbb{R} : x > 0\} 
S \Coloneqq \{x \in \mathbb{R} : x > 0\}

Defining a mathematical function.

f(x)∷=x2+1f(x) \Coloneqq \sqrt{x^2 + 1} 
f(x) \Coloneqq \sqrt{x^2 + 1}