# Definition:Set Union/Family of Sets

(Redirected from Definition:Union of Family)

## Definition

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:

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

### In the context of the Universal Set

In treatments of set theory in which the concept of the universal set is recognised, this can be expressed as follows.

Let $\mathbb U$ be a universal set.

Let $I$ be an indexing set.

Let $\family {S_i}_{i \mathop \in I}$ be an indexed family of subsets of $\mathbb U$.

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

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

### Subsets of General Set

This definition is the same when the universal set $\mathbb U$ is replaced by any set $X$, which may or may not be a universal set:

Let $\family {S_i}_{i \mathop \in I}$ be an indexed family of subsets of a set $X$.

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

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

where $i$ is a dummy variable.

## Union of Family of Two Sets

Let $I = \set {\alpha, \beta}$ be an indexing set containing exactly two elements.

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

From the definition of the union of $S_i$:

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

it follows that:

$\ds \bigcup \set {S_\alpha, S_\beta} := S_\alpha \cup S_\beta$

## Also denoted as

The set $\ds \bigcup_{i \mathop \in I} S_i$ can also be seen denoted as:

$\ds \bigcup_I S_i$

or, if the indexing set is clear from context:

$\ds \bigcup_i S_i$

However, on this website it is recommended that the full form is used.

## Examples

### Example: $\size {y - 1} < n$ and $\size {y+1} > \dfrac 1 n$

Let $I$ be the indexing set $I = \set {1, 2, 3, \ldots}$

Let $\family {T_n}$ be the indexed family of subsets of the set of real numbers $\R$, defined as:

$T_n = \set {y: \size {y - 1} < n \land \size {y + 1} > \dfrac 1 n}$

Then:

$\ds \bigcup_{n \mathop \in I} T_n = \R \setminus \set {-1}$

## Also see

• Results about set unions can be found here.