# Symbols:Set Theory/Set Intersection

## Set Intersection

$\cap$

$S \cap T$ denotes the intersection of $S$ and $T$.

That is, $S \cap T$ is defined to be the set containing all the elements that are in both the sets $S$ and $T$:

$S \cap T := \set {x: x \in S \land x \in T}$

The $\LaTeX$ code for $\cap$ is \cap .

### Set of Sets

$\bigcap$

Let $\Bbb S$ be a set of sets.

The intersection of $\Bbb S$ is:

$\bigcap \Bbb S := \set {x: \forall S \in \Bbb S: x \in S}$

That is, the set of all objects that are elements of all the elements of $\Bbb S$.

Thus:

$\bigcap \set {S, T} := S \cap T$

The $\LaTeX$ code for $\bigcap$ is \bigcap .

### Family of Sets

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

Let $I$ be an indexing set.

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

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

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

The $\LaTeX$ code for $\ds \bigcap_{i \mathop \in I} S_i$ is \ds \bigcap_{i \mathop \in I} S_i .