PlanetMath: intersection

(more info)

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intersection

(Definition)

The intersection of two sets and is the set that contains all the elements such that $x \in A$ and $x \in B$ . The intersection of and is written as $A \cap B$ . The following Venn diagram illustrates the intersection of two sets and :

$\begin{pspicture}(0,0)(6,4) \pscircle[fillstyle=vlines,hatchcolor=red,hatchwidth... ...(3,2){$A\cap B$} \rput(5,2){$B$} \rput(0,0){$.$} \rput(6,4){$.$} \end{pspicture}$

Example. If $A=\{1,2,3,4,5\}$ and $B=\{1,3,5,7,9\}$ then $A\cap B=\{1,3,5\}$ .

We can also define the intersection of an arbitrary number of sets. If $\{A_j\}_{j\in J}$ is a family of sets, we define the intersection of all them, denoted $\bigcap_{j\in J} A_j$ , as the set consisting of those elements belonging to every set :

$\displaystyle \bigcap_{j\in J} A_ j = \{x: x\in A_j$ for all $\displaystyle j\in J \}.$

A set intersects, or meets, a set if $U\cap V$ is non-empty.

Some elementary properties of $\cap$ are

(idempotency) $A\cap A = A$ ,
(commutativity) $A\cap B=B\cap A$ ,
(associativity) $A\cap (B\cap C) = (A\cap B)\cap C$ ,
$A\cap A^\complement = \varnothing$ , where $A^\complement$ is the complement of in some fixed universe .

Remark. What is $\bigcap_{j\in J} A_j$ when $J=\varnothing$ ? In other words, what is the intersection of an empty family of sets? First note that if $I\subseteq J$ , then

$\displaystyle \bigcap_{j\in J} A_j \subseteq \bigcap_{i\in I} A_i.$

This leads the conclusion that the intersection of an empty family of sets should be as large as possible. How large should it be? The answer is that we want it just large enough so that all sets that are under scrutiny are its subsets. Therefore, if we fix a universe