In mathematics, the law of trichotomy states that every

real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every r ...

is either positive, negative, or zero.Trichotomy Law
at

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More generally, a

binary relation In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over sets and is a new set of ordered pairs consisting of elements in and i ...

''R'' on a set ''X'' is trichotomous if for all ''x'' and ''y'' in ''X'', exactly one of ''xRy'', ''yRx'' and ''x''=''y'' holds. Writing ''R'' as <, this is stated in formal logic as: :

\forall x \in X \, \forall y \in X \, (x < y  \, \land \, \lnot(y < x) \, \land \, \lnot(x = y) \, \lor \, \lnot(x < y) \, \land \,       y < x  \, \land \, \lnot(x = y) \, \lor \, \lnot(x < y) \, \land \, \lnot(y < x) \, \land \,       x = y) \,.

Properties

* A relation is trichotomous if, and only if, it is asymmetric and

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. * If a trichotomous relation is also transitive, then it is a

strict total order In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( reflexive ...

; this is a special case of a strict weak order.

Examples

* On the set ''X'' = , the relation ''R'' = is transitive and trichotomous, and hence a strict

total order In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( reflexi ...

. * On the same set, the cyclic relation ''R'' = is trichotomous, but not transitive; it is even

antitransitive In mathematics, intransitivity (sometimes called nontransitivity) is a property of binary relations that are not transitive relations. This may include any relation that is not transitive, or the stronger property of antitransitivity, which descr ...

Trichotomy on numbers

A law of trichotomy on some set ''X'' of numbers usually expresses that some tacitly given ordering relation on ''X'' is a trichotomous one. An example is the law "For arbitrary real numbers ''x'' and ''y'', exactly one of ''x'' < ''y'', ''y'' < ''x'', or ''x'' = ''y'' applies"; some authors even fix ''y'' to be zero, relying on the real number's additive

linearly ordered group In mathematics, specifically abstract algebra, a linearly ordered or totally ordered group is a group ''G'' equipped with a total order "≤" that is ''translation-invariant''. This may have different meanings. We say that (''G'', ≤) is a: * le ...

structure. The latter is a

group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...

equipped with a trichotomous order. In classical logic, this axiom of trichotomy holds for ordinary comparison between real numbers and therefore also for comparisons between

integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language o ...

s and between

rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ratio ...

s. The law does not hold in general in

intuitionistic logic Intuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic by more closely mirroring the notion of constructive proof. In particular, systems o ...

. In

Zermelo–Fraenkel set theory In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such a ...

and Bernays set theory, the law of trichotomy holds between the

cardinal number In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. Th ...

s of well-orderable sets even without the

axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...

. If the axiom of choice holds, then trichotomy holds between arbitrary cardinal numbers (because they are all well-orderable in that case).

References

{{reflist Order theory Binary relations 3 (number)

Properties

Examples

Trichotomy on numbers

See also

References