In set theory, the union of sets is a fundamental operation, represented by the symbol ∪. It denotes the combination of elements from two or more sets.
To include this symbol in a scientific document using LaTeX, use the \cup command. Below is an example demonstrating its usage.
\[ A \cup B = \{x : x\in A \; or \; x\in B \} \]
\[ A \cup \phi \; or \; A\cup \{\} = A \]
\[ (A_1 \cup A_2 \cup A_3 \cup \cdots A_n)^c=A_1^c \cap A_2^c \cap A_3^c\cdots \cap A_n^c \]
\[ A\cup (B\cup C)=(A\cup B)\cup C \]
Output :
Big union symbol
When working with multiple sets, the union notation resembles summation notation but uses ∪ instead of ∑. In LaTeX, you can write.
\[ \cup_{i=1}^{n}F_{i} \]
Output :
However, this does not properly position the limits. To correctly display them, use the \bigcup command.
\[ \bigcup_{i=1}^{n}F_{i} \]
Output :
To correctly position upper and lower limits, use \limits with \bigcup. More details are provided below
\[ \bigcup\limits_{i=1}^{n}F_{i} \]
Output :
Among these options, the last two methods are the most effective for handling unions over multiple sets.
Disjoint union symbol
In LaTeX, there are three primary ways to represent a disjoint union operation in a document.
Using a Dot Over the Symbol
One method is to place a dot above the symbol. This can be done using the \dot{\cup} command.
\[ S_{1} \dot\cup S_{2} \]
Output :
Using a Plus Sign
Another method is to use the \uplus command, which adds a small plus sign to the symbol, providing a distinct visual representation.
\[ S_{1} \uplus S_{2} \]
Output :
Four-corner U-shaped
Alternatively, you can use a four-corner U-shaped symbol with the \sqcup command.
\[ S_{1} \sqcup S_{2} \]
Output :
In addition to these default methods, you can also install an external LaTeX package to access more advanced notations.
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