Learn how to write the union symbol (∪) in LaTeX using the \cup command. This guide covers set union operations, big union notation, and complete set theory symbols for mathematical documents.
To write the union symbol (∪) in LaTeX:
$A \cup B$ % Produces: A ∪ BFor big union (multiple sets), use: \bigcup_{i=1}^{n} A_i
The \cup command creates the union symbol (∪) in LaTeX, used in set theory to denote the union of two or more sets. The union A ∪ B contains all elements that are in A, in B, or in both.
% Simple union of two sets
$A \cup B$
% Union with set notation
$\{1, 2\} \cup \{2, 3\} = \{1, 2, 3\}$
% Multiple unions
$A \cup B \cup C$
% In display math mode
\[ A \cup B \cup C \cup D \]Output:
A ∪ B
{1, 2} ∪ {2, 3} = {1, 2, 3}
A ∪ B ∪ C
Set Theory Definition: The union of sets A and B, written A ∪ B, is the set of all elements that belong to A, to B, or to both. Formally: A ∪ B = {x | x ∈ A or x ∈ B}
% Union of intervals
$(0, 1) \cup (1, 2)$
% Union of number sets
$\mathbb{N} \cup \{0\} = \mathbb{N}_0$ % Natural numbers with zero
% Empty set union
$A \cup \emptyset = A$
% Universal set property
$A \cup U = U$ where $U$ is the universal setThe \bigcup command creates a large union symbol used for union operations over multiple sets, often with index notation.
% Big union with subscript
$\bigcup_{i=1}^{n} A_i$
% Union of infinite family of sets
$\bigcup_{i \in I} A_i$
% Display style (larger)
\[
\bigcup_{i=1}^{\infty} A_i = A_1 \cup A_2 \cup A_3 \cup \cdots
\]
% Without limits (inline style)
$\bigcup A_i$Output:
⋃i=1n Ai
⋃i∈I Ai
Display vs Inline Style: In display math mode \[ \], limits appear above and below the union symbol. In inline math $ $, they appear as subscripts and superscripts. Force display style inline with \displaystyle.
% Union of indexed family
\[
\bigcup_{i=1}^{n} [i, i+1] = [1, n+1]
\]
% Countable union
$\bigcup_{n=1}^{\infty} A_n$ represents a countable union.
% Arbitrary union
$\bigcup_{\alpha \in \Lambda} A_{\alpha}$ for index set $\Lambda$
% Disjoint union notation (sometimes)
$\bigsqcup_{i \in I} A_i$ % Uses \bigsqcup for disjoint unionThe union symbol is part of a family of set operation symbols in LaTeX:
| Symbol | Command | Name | Meaning |
|---|---|---|---|
| ∪ | \cup |
Union | A ∪ B (elements in A or B) |
| ⋃ | \bigcup |
Big Union | Union of multiple sets |
| ∩ | \cap |
Intersection | A ∩ B (elements in both A and B) |
| ⋂ | \bigcap |
Big Intersection | Intersection of multiple sets |
| ∖ or \ | \setminus |
Set Difference | A \ B (elements in A but not B) |
| △ | \triangle |
Symmetric Difference | A △ B (in A or B but not both) |
| ⊆ | \subseteq |
Subset or Equal | A ⊆ B (A is subset of B) |
| ⊂ | \subset |
Proper Subset | A ⊂ B (A is proper subset of B) |
| ∈ | \in |
Element Of | x ∈ A (x is element of A) |
| ∅ | \emptyset |
Empty Set | Set with no elements |
% Define sets
Let $A = \{1, 2, 3, 4\}$ and $B = \{3, 4, 5, 6\}$.
% Union
\[ A \cup B = \{1, 2, 3, 4, 5, 6\} \]
% Intersection
\[ A \cap B = \{3, 4\} \]
% Set difference
\[ A \setminus B = \{1, 2\} \]
\[ B \setminus A = \{5, 6\} \]% Overlapping intervals
\[ [1, 3] \cup [2, 4] = [1, 4] \]
% Disjoint intervals
\[ [1, 2] \cup [3, 4] = [1, 2] \cup [3, 4] \]
% (Cannot be simplified)
% Infinite union of intervals
\[
\bigcup_{n=1}^{\infty} \left[\frac{1}{n+1}, \frac{1}{n}\right] = (0, 1]
\]% Combining number sets
\[ \mathbb{Q} \cup \mathbb{I} = \mathbb{R} \]
% (Rationals union Irrationals = Reals)
% Natural numbers with zero
\[ \mathbb{N} \cup \{0\} = \mathbb{N}_0 = \mathbb{W} \]
% (Natural numbers union zero = Whole numbers)
% Complex numbers decomposition
\[ \mathbb{R} \cup i\mathbb{R} \subset \mathbb{C} \]% Union of nested sets
Let $A_n = \{1, 2, \ldots, n\}$ for each $n \in \mathbb{N}$.
\[
\bigcup_{n=1}^{\infty} A_n = \mathbb{N}
\]
% Union of balls (topology)
\[
\bigcup_{r > 0} B(x, r) = X
\]
where $B(x, r)$ is an open ball of radius $r$ centered at $x$.% Union of events
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
% Mutually exclusive events
If $A \cap B = \emptyset$, then:
\[ P(A \cup B) = P(A) + P(B) \]
% Countable union (sigma-additivity)
For disjoint events $A_1, A_2, \ldots$:
\[
P\left(\bigcup_{i=1}^{\infty} A_i\right) = \sum_{i=1}^{\infty} P(A_i)
\]While LaTeX doesn't draw Venn diagrams automatically, you can describe union operations clearly:
% Describing union visually
The shaded region in the Venn diagram represents $A \cup B$:
all areas covered by circle $A$, circle $B$, or both.
% With TikZ (advanced - requires \usepackage{tikz})
\begin{tikzpicture}
\draw (0,0) circle (1cm) node {$A$};
\draw (1,0) circle (1cm) node {$B$};
\node at (0.5, -1.5) {$A \cup B$};
\end{tikzpicture}Visual Representation: In a Venn diagram, A ∪ B is represented by shading both circles entirely. This includes the overlapping region (A ∩ B) and the unique parts of each set.
% Commutative property
\[ A \cup B = B \cup A \]
% Associative property
\[ (A \cup B) \cup C = A \cup (B \cup C) \]
% Idempotent property
\[ A \cup A = A \]
% Identity element (empty set)
\[ A \cup \emptyset = A \]
% Domination (universal set)
\[ A \cup U = U \]% De Morgan's first law
\[ \overline{A \cup B} = \overline{A} \cap \overline{B} \]
% De Morgan's second law
\[ \overline{A \cap B} = \overline{A} \cup \overline{B} \]
% Generalized De Morgan's law
\[
\overline{\bigcup_{i \in I} A_i} = \bigcap_{i \in I} \overline{A_i}
\]% Union distributes over intersection
\[ A \cup (B \cap C) = (A \cup B) \cap (A \cup C) \]
% Intersection distributes over union
\[ A \cap (B \cup C) = (A \cap B) \cup (A \cap C) \]Remember: \cup (∪) is union (elements in A or B), while \cap (∩) is intersection (elements in A and B). The symbols look similar but have opposite meanings!
% WRONG - using intersection instead of union
$\{1, 2\} \cap \{2, 3\} = \{1, 2, 3\}$ % Incorrect!
% CORRECT - union includes all elements
$\{1, 2\} \cup \{2, 3\} = \{1, 2, 3\}$ % Correct!
% Intersection is only common elements
$\{1, 2\} \cap \{2, 3\} = \{2\}$ % Correct intersection% WRONG - causes error
The union of A and B is A \cup B.
% CORRECT - use math mode
The union of A and B is $A \cup B$.% WRONG - missing underscore for subscript
$\bigcup i=1^n A_i$
% CORRECT - use underscore for limits
$\bigcup_{i=1}^{n} A_i$Use the \cup command in math mode: $A \cup B$ produces A ∪ B. For large union of multiple sets, use \bigcup.
\cup creates a small union symbol for binary operations (A ∪ B). \bigcup creates a large symbol used with limits for union of many sets: \bigcup_{i=1}^{n} A_i.
In LaTeX (based on TeX), use $A \cup B$. The command is \cup (short for "cup" shape). Always use math mode with dollar signs or equation environments.
There's no universal keyboard shortcut. In LaTeX, always type \cup. Some editors provide autocomplete: type \cup and press Tab. The Unicode character is ∪ (U+222A) but using LaTeX commands is recommended.
Use \limits or \nolimits: $\bigcup\limits_{i=1}^{n} A_i$ forces limits above/below even in inline math. Or use \displaystyle: $\displaystyle\bigcup_{i=1}^{n} A_i$.
No special package required! \cup and \bigcup are built into standard LaTeX. They work in any math mode without additional packages.
Use \sqcup for disjoint union: $A \sqcup B$ produces A ⊔ B. For big disjoint union, use \bigsqcup: $\bigsqcup_{i=1}^{n} A_i$.
No, \cup only works in math mode. Wrap it in dollar signs when using in text: "The set $A \cup B$ contains..." or use \ensuremath{\cup} for dual-mode usage.
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