Intersection

In set theory, the intersection of two sets is the set of elements that they have in common:

$A\cap B=\{x:x\in A\wedge x\in B\},\,$

where $\wedge$ denotes logical and.

Properties

The intersection operation is:

• associative : $(A\cap B)\cap C=A\cap (B\cap C)$;
• commutative : $A\cap B=B\cap A$.

General intersections

Finite intersections

The intersection of any finite number of sets may be defined inductively, as

$\bigcap _{i=1}^{n}X_{i}=X_{1}\cap (X_{2}\cap (X_{3}\cap (\cdots X_{n})\cdots ))).\,$

Infinite intersections

The intersection of a general family of sets X_λ as λ ranges over a general index set Λ may be written in similar notation as

$\bigcap _{\lambda \in \Lambda }X_{\lambda }=\{x:\forall \lambda \in \Lambda ,~x\in X_{\lambda }\}.\,$

We may drop the indexing notation and define the intersection of a set to be the set of elements contained in all the elements of that set:

$\bigcap X=\{x:\forall Y\in X,~x\in Y\}.\,$

In this notation the intersection of two sets A and B may be expressed as

$A\cap B=\bigcap \{A,B\}.\,$

The correct definition of the intersection of the empty set needs careful consideration.

See also

• Union

References

• Paul Halmos (1960). Naive set theory. Van Nostrand Reinhold.  Section 4.
• Keith J. Devlin (1979). Fundamentals of Contemporary Set Theory. Springer-Verlag, 6,11. ISBN 0-387-90441-7.