# Axiom:Axiom of Dependent Choice

## Axiom

### Left-Total Form

Let $\RR$ be a binary relation on a non-empty set $S$.

Suppose that:

$\forall a \in S: \exists b \in S: a \mathrel \RR b$

that is, that $\RR$ is a left-total relation (specifically a serial relation).

The axiom of dependent choice states that there exists a sequence $\sequence {x_n}_{n \mathop \in \N}$ in $S$ such that:

$\forall n \in \N: x_n \mathrel \RR x_{n + 1}$

### Right-Total Form

Let $\mathcal R$ be a binary relation on a non-empty set $S$.

Suppose that:

$\forall a \in S: \exists b \in S: b \mathrel {\mathcal R} a$

that is, that $\mathcal R$ is a right-total relation.

The axiom of dependent choice states that there exists a sequence $\sequence {x_n}_{n \mathop \in \N}$ in $S$ such that:

$\forall n \in \N: x_{n + 1} \mathrel {\mathcal R} x_n$

## Also known as

Some sources call this the Axiom of Dependent Choices, reflecting the infinitely many choices made.

This axiom can be abbreviated ADC or simply DC.