⊿ Point. {R} Glossary. ◢ Keyword: C. ◥ University. {q} PhD. {tr} Training. ⚫ UK. ↂ EndNote. ✊ Harvey (2009) Comparability. Property of elements related by inequalities 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[edit] A binary relation on a set is by definition any subset of Given is written if and only if in which case is said to be related to by An element is said to be -comparable, or comparable (with respect to ), to an element if or Often, a symbol indicating comparison, such as (or and many others) is used instead of is written in place of which is why the term "comparable" is used.
Comparability with respect to induces a canonical binary relation on ; specifically, the comparability relation induced by is defined to be the set of all pairs such that is comparable to ; that is, such that at least one of and is true. Induced by is incomparable to that is, such that neither nor is true. If the symbol is used in place of then comparability with respect to.