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 : $$A \cap B = B \cap A$$.

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 Xλ 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.