# Axiom:Axiom of Dependent Choice

## Axiom

### Left-Total Form

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

Suppose that:

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

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

The axiom of dependent choice states that there exists a sequence $\left\langle{x_n}\right\rangle_{n \in \N}$ in $S$ such that:

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

### 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 $\left\langle{x_n}\right\rangle_{n \in \N}$ 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.