Definition:Class Intersection

Definition

Let $A$ and $B$ be two classes.

The (class) intersection $A \cap B$ of $A$ and $B$ is defined as the class of all sets $x$ such that $x \in A$ and $x \in B$:

$x \in A \cap B \iff x \in A \land x \in B$

or:

$A \cap B = \set {x: x \in A \land x \in B}$

Class of Sets

Let $A$ be a class.

The intersection of $A$ is:

$\bigcap A := \set {x: \forall y \in A: x \in y}$

That is, the class of all objects which belong to all the elements of $A$.

Also see

• Definition:Set Intersection, the usual presentation of this concept in set theory

• Definition:Class Union

• Results about class intersections can be found here.

Internationalization

Intersection is translated:

In German:	durchschnitt	(literally: (act of) cutting)
In Dutch:	doorsnede

Sources

• 2002: Thomas Jech: Set Theory (3rd ed.) ... (previous) ... (next): Chapter $1$: Classes
• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $2$: Some Basics of Class-Set Theory: $\S 5$ The union axiom: Boolean operations $(2)$