# Definition:Set Intersection/Family of Sets

## 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 intersection of $\family {S_i}$ is defined as:

$\displaystyle \bigcap_{i \mathop \in I} S_i := \set {x: \forall i \in I: x \in S_i}$

### 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 intersection of $\family {S_i}$ is defined and denoted as:

$\displaystyle \bigcap_{i \mathop \in I} S_i := \set {x \in \mathbb U: \forall i \in I: x \in S_i}$

### 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 intersection of $\family {S_i}$ is defined as:

$\displaystyle \bigcap_{i \mathop \in I} S_i := \set {x \in X: \exists i \in I: x \in S_i}$

## Intersection 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 intersection of $S_i$:

$\displaystyle \bigcap_{i \mathop \in I} S_i := \set {x: \forall i \in I: x \in S_i}$

it follows that:

$\displaystyle \bigcap \set {S_\alpha, S_\beta} := S_\alpha \cap S_\beta$

## Also denoted as

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

$\displaystyle \bigcap_I S_i$

or, if the indexing set is clear from context:

$\displaystyle \bigcap_i S_i$

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