mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, 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 duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...

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

MathWorld ''MathWorld'' is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by and licensed to Wolfram Research, Inc. and was partially funded by the National Science Foundation's National Science ...

More generally, a

binary relation In mathematics, a binary relation associates some elements of one Set (mathematics), set called the ''domain'' with some elements of another set called the ''codomain''. Precisely, a binary relation over sets X and Y is a set of ordered pairs ...

''R'' on a

set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...

''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) \,.

With this definition, the law of trichotomy states that < is a trichotomous relation on the set of real numbers. In other words, if ''x'' and ''y'' are real numbers, then exactly one of the following must be true: ''x''<''y'', ''x''=''y'', ''y''<''x''.

Properties

* A relation is trichotomous if, and only if, it is asymmetric and connected. * If a trichotomous relation is also transitive, then it is a

strict total order In mathematics, a total order 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 ( ref ...

; 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 order 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 ( re ...

. * 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. That is, we can find three values a, b, and c where the transitive condition does not hold. Antitransitivity ...

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 fix ''y'' to be zero, relying on the real number's

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

structure for addition. In classical logic, this axiom of trichotomy holds for ordinary comparisons between real numbers and therefore also for comparisons between

integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...

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 (for example, The set of all ...

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 ...

. 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 suc ...

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

cardinal number In mathematics, a cardinal number, or cardinal for short, is what is commonly called the number of elements of a set. In the case of a finite set, its cardinal number, or cardinality is therefore a natural number. For dealing with the cas ...

s of well-orderable sets, but not necessarily for all cardinal numbers. 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 Properties of binary relations 3 (number)

Properties

Examples

Trichotomy on numbers

See also

References