Symbols:Set Theory/Set Union

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Set Union

$\cup$


Let $S$ and $T$ be sets.


The (set) union of $S$ and $T$ is the set $S \cup T$, which consists of all the elements which are contained in either (or both) of $S$ and $T$:

$x \in S \cup T \iff x \in S \lor x \in T$


The $\LaTeX$ code for \(\cup\) is \cup .


Set of Sets

$\bigcup$


Let $\mathbb S$ be a set of sets.

The union of $\mathbb S$ is:

$\bigcup \mathbb S := \set {x: \exists X \in \mathbb S: x \in X}$

That is, the set of all elements of all elements of $\mathbb S$.


Thus the general union of two sets can be defined as:

$\bigcup \set {S, T} = S \cup T$


The $\LaTeX$ code for \(\bigcup\) is \bigcup .


Family of Sets

$\ds \bigcup_{i \mathop \in I} S_i$


Let $I$ be an indexing set.

Let $\family {S_i}_{i \mathop \in I}$ be a family of sets indexed by $I$.


Then the union of $\family {S_i}$ is defined as:

$\ds \bigcup_{i \mathop \in I} S_i := \set {x: \exists i \in I: x \in S_i}$


The $\LaTeX$ code for \(\ds \bigcup_{i \mathop \in I} S_i\) is \ds \bigcup_{i \mathop \in I} S_i .


Also see


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