Definition:Set Intersection

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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:

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


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:

VennDiagramSetIntersection.png


Examples

Example: $2$ Arbitrarily Chosen Sets

Let:

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

Then:

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


Example: $3$ Arbitrarily Chosen Sets

Let:

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

Then:

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


Example: Blue-Eyed British People

Let:

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

Then:

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


Example: Overlapping Integer Sets

Let:

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

Then:

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

and so is finite.


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.


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 $1908$.


Sources