# Category:Examples of Use of Axiom of Dependent Choice

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This category contains examples of use of Axiom:Axiom of Dependent Choice.

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

Suppose that:

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

that is, that $\RR$ 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:

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

## Pages in category "Examples of Use of Axiom of Dependent Choice"

The following 14 pages are in this category, out of 14 total.

### I

- Infinite Sequence Property of Strictly Well-Founded Relation
- Infinite Sequence Property of Strictly Well-Founded Relation/Reverse Implication
- Infinite Sequence Property of Strictly Well-Founded Relation/Reverse Implication/Proof 1
- Infinite Sequence Property of Strictly Well-Founded Relation/Reverse Implication/Proof 2
- Infinite Sequence Property of Well-Founded Relation
- Infinite Sequence Property of Well-Founded Relation/Reverse Implication
- Infinite Sequence Property of Well-Founded Relation/Reverse Implication/Proof 1
- Infinite Sequence Property of Well-Founded Relation/Reverse Implication/Proof 2
- Infinite Set has Countably Infinite Subset/Intuitive Proof