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:

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


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:

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


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:

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

where $i$ is a dummy variable.


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 $S_i$:

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

it follows that:

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


Also denoted as

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

$\ds \bigcap_I S_i$

or, if the indexing set is clear from context:

$\ds \bigcap_i S_i$


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:

$\ds \bigcap_{n \mathop \in I} T_n = \openint 0 2$


Also see

  • Results about set intersections can be found here.


Sources

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