Definition:Diagonal Relation/Class Theory

Definition

Let $V$ be a basic universe.

The diagonal relation on $V$ is the relation $\Delta_V$ on $V$ defined as:

$\Delta_V = \set {\tuple {x, x}: x \in V}$

Alternatively:

$\Delta_V = \set {\tuple {x, y}: x, y \in V: x = y}$

Also known as

The diagonal relation can also be referred to as:

the equality relation or relation of equality

the identity relation.

It is also referred to it as:

the diagonal set or diagonal class

the diagonal subset or diagonal subclass.

Some sources call it just the diagonal.

However, $\mathsf{Pr} \infty \mathsf{fWiki}$'s position is that it can be useful to retain the emphasis that it is indeed a relation.

Also see

• Definition:Equals
• Diagonal Relation is Equivalence

• Definition:Identity Mapping

• Results about the diagonal relation can be found here.

Sources

• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $4$: Superinduction, Well Ordering and Choice: Part $\text I$ -- Superinduction and Well Ordering: $\S 1$ Introduction to well ordering