# Definition:Set Intersection

## Definition

Let $S$ and $T$ be any two sets.

The (set) intersection of $S$ and $T$ is written $S \cap T$.

It means the set which consists of all the elements which are contained in both of $S$ and $T$:

$x \in S \cap T \iff x \in S \land x \in T$

or, more formally:

$A = S \cap T \iff \forall z: \paren {z \in A \iff z \in S \land z \in T}$

We can write:

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

and can voice it $S$ intersect $T$.

It can be seen that, in this form, $\cap$ is a binary operation which acts on sets.

One often says that two sets intersect if and only if they have non-empty intersection.

### Set of Sets

Let $\Bbb S$ be a set of sets

The intersection of $\Bbb S$ is:

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

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

Thus:

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

### Family of Sets

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:

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

### Countable Intersection

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

Let $\left\langle{S_n}\right\rangle_{n \mathop \in \N}$ be a sequence in $\mathbb S$.

Let $S$ be the intersection of $\left\langle{S_n}\right\rangle_{n \mathop \in \N}$:

$\displaystyle S = \bigcap_{n \mathop \in \N} S_n$

Then $S$ is a countable intersection of sets in $\mathbb S$.

### Finite Intersection

Let $S = S_1 \cap S_2 \cap \ldots \cap S_n$.

Then:

$\displaystyle S = \bigcap_{i \mathop \in \N^*_n} S_i := \set {x: \forall i \in \N^*_n: x \in S_i}$

where $\N^*_n = \set {1, 2, 3, \ldots, n}$.

If it is clear from the context that $i \in \N^*_n$, we can also write $\displaystyle \bigcap_{\N^*_n} S_i$.

## Illustration by Venn Diagram

The intersection $S \cap T$ of the two sets $S$ and $T$ is illustrated in the following Venn diagram by the red area: ## Also known as

The intersection of sets is also known as the product, but this is usually considered old-fashioned nowadays.

The term meet can also be seen, but this is usually reserved for meet in order theory.

Some authors use the notation $S \ T$ or $S \cdot T$ for $S \cap T$, but this is non-standard and can be confusing.

## Examples

### Example: $2$ Arbitrarily Chosen Sets

Let:

 $\displaystyle S$ $=$ $\displaystyle \set {a, b, c}$ $\displaystyle T$ $=$ $\displaystyle \set {c, e, f, b}$

Then:

$S \cap T = \set {b, c}$

### Example: $2$ Arbitrarily Chosen Sets of Complex Numbers: $1$

Let:

 $\displaystyle A$ $=$ $\displaystyle \set {3, -i, 4, 2 + i, 5}$ $\displaystyle B$ $=$ $\displaystyle \set {-i, 0, -1, 2 + i}$

Then:

$A \cap B = \set {-i, 2 + i}$

### Example: $2$ Arbitrarily Chosen Sets of Complex Numbers: $2$

Let:

 $\displaystyle A$ $=$ $\displaystyle \set {3, -i, 4, 2 + i, 5}$ $\displaystyle C$ $=$ $\displaystyle \set {-\sqrt 2 i, \dfrac 1 2, 3}$

Then:

$A \cap C = \set 3$

### Example: $3$ Arbitrarily Chosen Sets

Let:

 $\displaystyle A_1$ $=$ $\displaystyle \set {1, 2, 3, 4}$ $\displaystyle A_2$ $=$ $\displaystyle \set {1, 2, 5}$ $\displaystyle A_3$ $=$ $\displaystyle \set {2, 4, 6, 8, 12}$

Then:

$A_1 \cap A_2 \cap A_3 = \set 2$

### Example: $3$ Arbitrarily Chosen Sets of Complex Numbers

Let:

 $\displaystyle A$ $=$ $\displaystyle \set {3, -i, 4, 2 + i, 5}$ $\displaystyle B$ $=$ $\displaystyle \set {-i, 0, -1, 2 + i}$ $\displaystyle C$ $=$ $\displaystyle \set {-\sqrt 2 i, \dfrac 1 2, 3}$

Then:

$B \cap C = \O$

and so it follows that:

$A \cap \paren {B \cap C} = \O$

### Example: $4$ Arbitrarily Chosen Sets of Complex Numbers

Let:

 $\displaystyle A$ $=$ $\displaystyle \set {1, i, -i}$ $\displaystyle B$ $=$ $\displaystyle \set {2, 1, -i}$ $\displaystyle C$ $=$ $\displaystyle \set {i, -1, 1 + i}$ $\displaystyle D$ $=$ $\displaystyle \set {0, -i, 1}$

Then:

$\paren {A \cup C} \cap \paren {B \cup D} = \set {1, -i}$

### Example: Blue-Eyed British People

Let:

 $\displaystyle B$ $=$ $\displaystyle \set {\text {British people} }$ $\displaystyle C$ $=$ $\displaystyle \set {\text {Blue-eyed people} }$

Then:

$B \cap C = \set {\text {Blue-eyed British people} }$

### Example: Overlapping Integer Sets

Let:

 $\displaystyle A$ $=$ $\displaystyle \set {x \in \Z: 2 \le x}$ $\displaystyle B$ $=$ $\displaystyle \set {x \in \Z: x \le 5}$

Then:

$A \cap B = \set {2, 3, 4, 5}$

and so is finite.

## Also see

• Results about set intersections can be found here.

## Internationalization

Intersection is translated:

 In German: durchschnitt (literally: (act of) cutting) In Dutch: doorsnede

## Historical Note

The concept of set intersection, or logical multiplication, was stated by Leibniz in his initial conception of symbolic logic.

The symbol $\cap$, informally known as cap, was first used by Hermann Günter Grassmann in Die Ausdehnungslehre from $1844$.

However, he was using it as a general operation symbol, not specialized for set intersection.

It was Giuseppe Peano who took this symbol and used it for intersection, in his $1888$ work Calcolo geometrico secondo l'Ausdehnungslehre di H. Grassmann.

Peano also created the large symbol $\bigcap$ for general intersection of more than two sets.

This appeared in his Formulario Mathematico, 5th ed. of $1908$.