In mathematics, two elements x and y of a set P are said to be comparable with respect to a binary relation ≤ if at least one of x ≤ y or y ≤ x is true. They are called incomparable if they are not comparable.

Rigorous definition

A binary relation is said to be is said to be , or ( which is why the term "comparable" is used.

; specifically, the is true. Similarly, the is true.

. is true.

Example

A totally ordered set is a partially ordered set in which any two elements are comparable. The Szpilrajn extension theorem states that every partial order is contained in a total order. Intuitively, the theorem says that any method of comparing elements that leaves some pairs incomparable can be extended in such a way that every pair becomes comparable.

Properties

Both of the relations and are symmetric and likewise for incomparability.

Comparability graphs

.

Classification

When classifying mathematical objects (e.g., topological spaces), two are said to be comparable when the objects that obey one criterion constitute a subset of the objects that obey the other, which is to say when they are comparable under the partial order ⊂. For example, the T1 and T2 criteria are comparable, while the T1 and sobriety criteria are not.

See also

• , a partial ordering in which incomparability is a transitive relation

References

External links