Definition:Relation/Class Theory

Definition

Let $V$ be a basic universe.

Let $A$ and $B$ be subclasses of $V$.

A relation $\RR$ is a subclass of the Cartesian product $A \times B$.

Note that in this context either or both of $A$ and $B$ can be $V$ itself.

Notation

If $\tuple {x, y}$ is an ordered pair such that $\tuple {x, y} \in \RR$, we use the notation:

$s \mathrel \RR t$

or:

$\map \RR {s, t}$

and can say:

$s$ bears $\RR$ to $t$

$s$ stands in the relation $\RR$ to $t$

If $\tuple {s, t} \notin \RR$, we can write: $s \not \mathrel \RR t$, that is, by drawing a line through the relation symbol.

Also see

• Results about relations can be found here.

Sources

• 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: $\S 8$ Relations