# Definition:Diagonal Relation/Class Theory

< Definition:Diagonal Relation(Redirected from Definition:Diagonal Relation (Class Theory))

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## 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 the **identity relation**.

It is also referred to it as:

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**.

## 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