Talk:Total relation

total
The example indicates that less than is not a total relation over the set of real numbers because "one can pick two equal numbers." If we are dealing with the SET of real numbers, then by definition, only one of each number can exist in that set. Therefore two equal numbers could never be picked, and less than could be a total relation. Can anyone clarify this?OKmisterWIKI (talk) 14:15, 19 March 2013 (UTC)


 * There is no requirement in that two different numbers have to be picked. One property of a total relation is that for every element a, that element will be related to itself under the relation. &mdash; Carl (CBM · talk) 14:47, 19 March 2013 (UTC)

Thank you. OKmisterWIKI (talk) 16:04, 19 March 2013 (UTC)

Connex vs Trichotomous
In this article, the notion of connex is introduced. In the article Binary relation, where a lot of notions are introduced regarding relations, this notion is called trichotomous. Should we take one of both names, or mention both on both places? BertSeghers (talk) 16:39, 12 August 2013 (UTC)

Challenge
The use of this namespace for a type of relation is not supported by a reference. In fact, references are available that use total relation in the sense of serial relation. If there is no objection, this article will be redirected to that one. For example see These references confirm standard usage departing from contents of this article. — Rgdboer (talk) 02:33, 8 April 2018 (UTC)
 * Gunther Schmidt (2013) Relational Mathematics, Definition 5.8, page 57, Cambridge University Press ISBN 9780511778810
 * Dahl/Damrath (2001) Mathematical Foundations of Computational Engineering, page 506

The Redirect has been made. Below is the old article (no supporting reference for Total relation). — Rgdboer (talk) 03:22, 13 April 2018 (UTC)

I've changed it to a disambiguation page, including a link to Connected relation. See there for reference documenting use of "total" for the property variously referred as "connected" or "connex". — Preceding unsigned comment added by Rzach (talk • contribs) 17:07, 21 April 2021 (UTC)

April 2018
In mathematics, a binary relation R over a set X is total or complete if for all a and b in X, a is related to b or b is related to a (or both).

In mathematical notation, this is
 * $$\forall a, b \in X(a R b \lor b R a)$$

Total relations are sometimes said to have comparability.

Examples
For example, "is less than or equal to" is a total relation over the set of real numbers, because for two numbers either the first is less than or equal to the second, or the second is less than or equal to the first. On the other hand, "is less than" is not a total relation, since one can pick two equal numbers, and then neither the first is less than the second, nor is the second less than the first. (But note that "is less than" is a weak order which gives rise to a total order, namely "is less than or equal to". The relationship between strict orders and weak orders is discussed at partially ordered set.)

The relation "is a subset of" is also not total because, for example, neither of the sets {1,2} and {3,4} is a subset of the other.

Properties and related notions
Totality implies reflexivity.

If a transitive relation is also total, it is a total preorder. If a partial order is also total, it is a total order.

A binary relation R over X is called connex if for all a and b in X such that a ≠ b, a is related to b or b is related to a (or both):
 * $$\forall a, b \in X(a R b \lor b R a\lor (a=b))$$

Connexity does not imply reflexivity. A strict partial order is a strict total order if and only if it is connex.