TeXipedia

bigcup

Represents a large union operator for combining multiple sets in mathematical notation.

Overview

Essential in set theory and mathematical logic for denoting the union of multiple sets or collections of elements. Commonly appears in:

  • Abstract algebra for combining multiple groups or sets
  • Database theory when merging result sets
  • Probability theory for uniting event spaces
  • Computer science for describing algorithm operations

Typically used with subscripts and superscripts to indicate the range or conditions of the union operation, and often appears in formal mathematical proofs and theoretical computer science literature.

Examples

Union of a sequence of sets indexed from 1 to n.

⋃i=1nAi=A1∪A2∪⋯∪An\bigcup_{i=1}^n A_i = A_1 \cup A_2 \cup \cdots \cup A_n 
\bigcup_{i=1}^n A_i = A_1 \cup A_2 \cup \cdots \cup A_n

Union of probability events in sample space.

P(⋃i=1nEi)≤∑i=1nP(Ei)P\left(\bigcup_{i=1}^n E_i\right) \leq \sum_{i=1}^n P(E_i) 
P\left(\bigcup_{i=1}^n E_i\right) \leq \sum_{i=1}^n P(E_i)

Union of vector spaces over a field.

V=⋃k=1mVkwhere Vk⊆RnV = \bigcup_{k=1}^m V_k \quad \text{where } V_k \subseteq \mathbb{R}^n 
V = \bigcup_{k=1}^m V_k \quad \text{where } V_k \subseteq \mathbb{R}^n