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intersection (Definition)

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


\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 $ A_j$:

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

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

Some elementary properties of $ \cap$ are

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 $ U$ such that all of the sets that are under discussion are subsets of $ U$, then we define the intersection of an empty family of sets (that are subsets of $ U$) to be $ U$.



"intersection" is owned by CWoo. [ full author list (5) | owner history (3) ]
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See Also: union, union, finite intersection property, empty set, product of left and right ideal

Other names:  intersects, meets
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Cross-references: subsets, conclusion, universe, fixed, complement, associativity, commutativity, idempotency, properties, Venn diagram, contains
There are 424 references to this entry.

This is version 17 of intersection, born on 2002-02-01, modified 2008-03-19.
Object id is 1630, canonical name is Intersection.
Accessed 21996 times total.

Classification:
AMS MSC03E99 (Mathematical logic and foundations :: Set theory :: Miscellaneous)

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