Symbols:Set Theory/Set Union

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 .

$\bigcup$

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 .

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

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 .