Definition:Set Intersection/Family of Sets

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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 $\left \langle {S_i} \right \rangle_{i \mathop \in I}$ be an indexed family of subsets of $\mathbb U$.


Then the intersection of $\left \langle {S_i} \right \rangle$ is defined as:

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


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 $\left \langle {S_i} \right \rangle_{i \mathop \in I}$ be an indexed family of subsets of a set $X$.


Then the intersection of $\left \langle {S_i} \right \rangle$ is defined as:

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


Intersection of Family of Two Sets

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

Let $\left \langle {S_i} \right \rangle_{i \mathop \in I}$ be a family of sets indexed by $I$.

From the definition of the intersection of $S_i$:

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

it follows that:

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


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.


Sources