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:

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


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.


Generalized Notation

Let $I$ be an indexing set.

Let $\left \langle {X_i} \right \rangle_{i \in I}$ be a family of subsets of a set $S$.

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

$\displaystyle \bigcap_{i \in I} X_i = \left\{{y: \exists i \in I: y \in X_i}\right\}$

This notation can also be used as $\displaystyle \bigcap_i X_i$ to be written $\displaystyle \bigcap_{i \in I} X_i$


The indexing set itself can be disposed of, as follows:

If $\Bbb S$ is a set of sets, then 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:

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


Countable Intersection

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

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


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


An alternative notation for the same concept is $\displaystyle \bigcap_{i=1}^n S_i$.


If $\Bbb S$ is a set of sets, then 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:

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


Illustration by Venn Diagram

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

VennDiagramSetIntersection.png


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.


Notes

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


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]


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

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