Definition:Set Intersection

From ProofWiki
Jump to navigation Jump to search

This page is about set intersection in the context of set theory. For other uses, see intersection.

Definition

Let $S$ and $T$ be 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:

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

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


Thus:

$\bigcap \set {S, T} := S \cap T$


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:

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


Countable Intersection

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

Let $\sequence {S_n}_{n \mathop \in \N}$ be a sequence in $\mathbb S$.

Let $S$ be the intersection of $\sequence {S_n}_{n \mathop \in \N}$:

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


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:

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

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 $\ds \bigcap_{\N^*_n} S_i$.


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


Also known as

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

Besides, it is used on $\mathsf{Pr} \infty \mathsf{fWiki}$ to mean a different concept.

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: $1$

Let:

\(\ds S\) \(=\) \(\ds \set {a, b, c}\)
\(\ds T\) \(=\) \(\ds \set {c, e, f, b}\)

Then:

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


Example: $2$ Arbitrarily Chosen Sets: $2$

Let:

\(\ds A\) \(=\) \(\ds \set {1, 2, 3, 4, 5, 6}\)
\(\ds B\) \(=\) \(\ds \set {1, 4, 5, 6, 7, 8}\)

Then:

$A \cap B = \set {1, 4, 5, 6}$


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

Let:

\(\ds A\) \(=\) \(\ds \set {3, -i, 4, 2 + i, 5}\)
\(\ds B\) \(=\) \(\ds \set {-i, 0, -1, 2 + i}\)

Then:

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


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

Let:

\(\ds A\) \(=\) \(\ds \set {3, -i, 4, 2 + i, 5}\)
\(\ds C\) \(=\) \(\ds \set {-\sqrt 2 i, \dfrac 1 2, 3}\)

Then:

$A \cap C = \set 3$


Example: $3$ Arbitrarily Chosen Sets

Let:

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

Then:

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


Example: $3$ Arbitrarily Chosen Sets of Complex Numbers

Let:

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

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:

\(\ds A\) \(=\) \(\ds \set {1, i, -i}\)
\(\ds B\) \(=\) \(\ds \set {2, 1, -i}\)
\(\ds C\) \(=\) \(\ds \set {i, -1, 1 + i}\)
\(\ds D\) \(=\) \(\ds \set {0, -i, 1}\)

Then:

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


Example: Blue-Eyed British People

Let:

\(\ds B\) \(=\) \(\ds \set {\text {British people} }\)
\(\ds C\) \(=\) \(\ds \set {\text {Blue-eyed people} }\)

Then:

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


Example: Overlapping Integer Sets

Let:

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

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 $1908$.


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Definition:Set_Intersection&oldid=722247"