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**** on**** a set P is**** by**** definition**** any subset**** R of**** P *tim**es** P.*** Given x,**** y *in*** P,**** x R y is**** written if**** and only**** if**** in**** which case**** x is**** said**** to**** be**** ' to y by R.** An**** element x *in*** P is**** said**** to**** be**** ', or , to an element y *in* P if**** x R y or**** y R x.**** **** Often,**** a symbol**** indicating**** comparison****,**** such**** as**** *,<*,**** (or *,*leq*****,,** ***,>,*,* *geq**,*** and many**** others****)**** is**** used**** instead of**** R,**** in**** which case**** x < y is**** written in**** place of**** x R y,**** which is**** why the term**** "comparable"**** is**** used****.**** Comparability with**** respect to**** R induces a canonical binary**** relation**** on**** P;**** specifically****,**** the **** induced by**** R is**** defined to**** be**** the set of**** all pairs such**** that**** x is**** comparable**** to**** y;**** that**** is****,**** such**** that**** at**** least one of**** x R y and y R x is**** true****.**** Similarly,**** the **** on**** P induced by**** R is**** defined to**** be**** the set of**** all pairs such**** that**** x is**** incomparable**** to**** y;**** that**** is****,**** such**** that**** neither x R y nor y R x is**** true****.**** If the symbol ,<, is used in place of ,\leq, then comparability with respect to ,<, is sometimes denoted by the symbol, and incomparability by the symbol. Thus, for any two elements x and y of a partially ordered set, exactly one of and 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, that is x is comparable to y if and only if y is comparable to x, and likewise for incomparability.

Comparability graphs

The comparability graph of a partially ordered set P has as vertices the elements of P and has as edges precisely those pairs { x, y } of elements for which.

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.