[summary:
A totally ordered set is a pair $~$(S, \le)$~$ of a set $~$S$~$ and a *total order* $~$\le$~$ on $~$S$~$, which is a binary relation that satisfies the following properties:

- For all $~$a, b \in S$~$, if $~$a \le b$~$ and $~$b \le a$~$, then $~$a = b$~$. (the antisymmetric property)
- For all $~$a, b, c \in S$~$, if $~$a \le b$~$ and $~$b \le c$~$, then $~$a \le c$~$. (the transitive property)
- For all $~$a, b \in S$~$, either $~$a \le b$~$ or $~$b \le a$~$, or both. (the [total_relation totality] property)]

A **totally ordered set** is a pair $~$(S, \le)$~$ of a set $~$S$~$ and a *total order* $~$\le$~$ on $~$S$~$, which is a [-binary_relation] that satisfies the following properties:

- For all $~$a, b \in S$~$, if $~$a \le b$~$ and $~$b \le a$~$, then $~$a = b$~$. (the antisymmetric property)
- For all $~$a, b, c \in S$~$, if $~$a \le b$~$ and $~$b \le c$~$, then $~$a \le c$~$. (the transitive property)
- For all $~$a, b \in S$~$, either $~$a \le b$~$ or $~$b \le a$~$, or both. (the [total_relation totality] property)

A totally ordered set is a special type of partially ordered set that satisfies the total property — in general, posets only satisfy the reflexive property, which is that $~$a \le a$~$ for all $~$a \in S$~$.

## Examples of totally ordered sets

The real numbers are a totally ordered set. So are any of the subsets of the real numbers, such as the rational numbers or the integers.

## Examples of not totally ordered sets

The complex numbers do not have a canonical total ordering, and especially not a total ordering that preserves all the properties of the ordering of the real numbers, although one can define a total ordering on them quite easily.

## Comments

Kevin Clancy

Correct me if I'm wrong, but isn't it idiosyncratic to define $~$\leq$~$ as a predicate rather than a relation? I know of at least three books that describe it as a relation: The Joy of Sets by Devlin, Principles of Mathematical Analysis by Rudin, and Introduction to Lattice and Order by Davey and Priestly.

Also, isn't $~$\leq$~$ called an order rather than a comparison?

I bring this up because I would like there to be consistency between this page and the Partially ordered set page. I think both pages should follow the conventions of mathematics.