Definition:Set Intersection

From ProofWiki
Jump to: navigation, search

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:

$A = S \cap T \iff \forall z: \left({z \in A \iff z \in S \land z \in T}\right)$


We can write:

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


For example, let $S = \left \{{1,2,3}\right\}$ and $T = \left \{{2,3,4}\right\}$. Then $S \cap T = \left \{{2,3}\right\}$.


It can be seen that $\cap$ is an operator.


One often says that two sets intersect iff they have non-empty intersection.


General Definition

Let $\Bbb S$ be a set of sets

The intersection of $\Bbb S$ is:

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

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


Thus:

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


Family of Sets

Let $I$ be an indexing set.

Let $\left \langle {S_i} \right \rangle_{i \mathop \in I}$ be a family of sets indexed by $I$.


Then the intersection of $\left \langle {S_i} \right \rangle$ is defined as:

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


Countable Intersection

Let $\Bbb S = \left\{{S_0, S_1, S_2, \ldots}\right\}$ be a set of a countably infinite number of sets.


Then:

$\displaystyle \bigcap \Bbb S := \bigcap_{i \mathop \in \N} S_i = \left\{{x: \forall i \in \N: x \in S_i}\right\}$

This can also be denoted:

$\displaystyle \bigcap_{i \mathop = 1}^\infty S_i$

but its usage is strongly discouraged.

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


Finite Intersection

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

Then:

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

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


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 red area in the following Venn diagram illustrates $S \cap T$:

VennDiagramSetIntersection.png


Historical Note

The symbol $\cap$, informally known as cap, was first used by Hermann Grassmann in Die Ausdehnungslehre from 1844. However, he was using it as a general operation symbol, not specialized for 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 edtion, 1908).[1]


Also denoted as

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

  • Intersection of Singleton, where it is shown that $\displaystyle \Bbb S = \left\{{S}\right\} \implies \bigcap \Bbb S = S$
  • Intersection of Empty Set, where it is shown (paradoxically) that $\displaystyle \Bbb S = \left\{{\varnothing}\right\} \implies \bigcap \Bbb S = \Bbb U$
  • Results about set intersections can be found here.


Internationalization

Intersection is translated:

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


References

  1. See Earliest Uses of Symbols of Set Theory and Logic in Jeff Miller's website Earliest Uses of Various Mathematical Symbols.


Sources