TeXipedia

mathcal

Provides stylized calligraphic letters commonly used for denoting special mathematical sets, operators, and formal script notation.

Overview

Transforms uppercase letters into elegant calligraphic forms, particularly valued in mathematical writing for distinguishing special sets and mathematical structures.

  • Frequently used to denote important mathematical sets (like Real numbers ℝ or Complex numbers ℂ)
  • Common in probability theory for denoting sigma-algebras and event spaces
  • Popular in advanced algebra for representing categories and special operators
  • Limited to uppercase letters A-Z (lowercase letters and numbers remain unchanged)
  • Often seen in academic papers and formal mathematical notation to add visual distinction to important mathematical objects

Examples

Using mathcal for denoting a mathematical set or collection

N={1,2,3,…}\mathcal{N} = \{1, 2, 3, \ldots\} 
\mathcal{N} = \{1, 2, 3, \ldots\}

Representing a probability space in statistics

P(A∣F)=P(A∩F)P(F)P(A \mid \mathcal{F}) = \frac{P(A \cap \mathcal{F})}{P(\mathcal{F})} 
P(A \mid \mathcal{F}) = \frac{P(A \cap \mathcal{F})}{P(\mathcal{F})}

Denoting a function space in analysis

f∈C∞(R)f \in \mathcal{C}^{\infty}(\mathbb{R}) 
f \in \mathcal{C}^{\infty}(\mathbb{R})