In mathematics, a binary relation R over a set X is total if it holds for all a and b in X that a is related to b or b is related to a (or both). Euclid, a famous Greek mathematician known as the father of geometry, is shown here in detail from The School of Athens by Raphael. ... In mathematics, the concept of binary relation, sometimes called dyadic relation, is exemplified by such ideas as is greater than and is equal to in arithmetic, or is congruent to in geometry, or is an element of or is a subset of in set theory. ... In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ...

In mathematical notation, this is Mathematical notation is used in mathematics, and throughout the physical sciences, engineering, and economics. ...

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 proper subset of" is also not total. In mathematics, especially order theory, a partially ordered set (or poset for short) is a set equipped with a partial order relation. ...

Total relations are sometimes said to have comparability.

A total order provides a common example of a total relation. In mathematics, a total order, linear order or simple order on a set X is any binary relation on X that is antisymmetric, transitive, and total. ...