TeXipedia

Cap

Represents the double intersection operation between sets, combining two or more sets to find common elements.

Overview

Serves as a fundamental notation in set theory and mathematical logic for denoting multiple set intersections. Commonly appears in:

  • Advanced set theory proofs and theorems
  • Mathematical analysis of overlapping collections
  • Discrete mathematics and computer science contexts
  • Formal logic expressions where multiple set intersections are needed

More visually distinctive than repeated use of standard intersection symbols, making complex expressions clearer and more readable.

Examples

Set intersection of three sets using the Cap symbol.

A⋒B⋒C={x:x∈A and x∈B and x∈C}A \Cap B \Cap C = \{x : x \in A \text{ and } x \in B \text{ and } x \in C\} 
A \Cap B \Cap C = \{x : x \in A \text{ and } x \in B \text{ and } x \in C\}

Multiple intersection of indexed sets using Cap notation.

⋂i=1nAi=A1⋒A2⋒⋯⋒An\bigcap_{i=1}^n A_i = A_1 \Cap A_2 \Cap \cdots \Cap A_n 
\bigcap_{i=1}^n A_i = A_1 \Cap A_2 \Cap \cdots \Cap A_n

Comparing regular intersection with Cap in set theory.

X∩Y≠X⋒Y⋒ZX \cap Y \neq X \Cap Y \Cap Z 
X \cap Y \neq X \Cap Y \Cap Z