Definition:Diagonal Relation/Class Theory

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Let $V$ be a basic universe.

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

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


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

Also known as

The diagonal relation can also be referred to as the equality relation or the identity relation.

It is also referred to it as:

the diagonal set or class
the diagonal subset or 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

  • Results about the diagonal relation can be found here.