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Law of trichotomy

In mathematics, the law of trichotomy states that every real number is either positive, negative, or zero.[1]

More generally, a binary relation 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:

Properties

Examples

  • On the set X = {a,b,c}, the relation R = { (a,b), (a,c), (b,c) } is transitive and trichotomous, and hence a strict total order.
  • On the same set, the cyclic relation R = { (a,b), (b,c), (c,a) } is trichotomous, but not transitive; it is even antitransitive.

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,[1] relying on the real number's additive linearly ordered group structure. The latter is a group 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 integers and between rational numbers.[clarification needed] The law does not hold in general in intuitionistic logic.[citation needed]

In Zermelo–Fraenkel set theory and Bernays set theory, the law of trichotomy holds between the cardinal numbers of well-orderable sets even without the axiom of choice. If the axiom of choice holds, then trichotomy holds between arbitrary cardinal numbers (because they are all well-orderable in that case).[4]

See also

References

  1. ^ a b Trichotomy Law at MathWorld
  2. ^ Jerrold E. Marsden & Michael J. Hoffman (1993) Elementary Classical Analysis, page 27, W. H. Freeman and Company ISBN 0-7167-2105-8
  3. ^ H.S. Bear (1997) An Introduction to Mathematical Analysis, page 11, Academic Press ISBN 0-12-083940-7
  4. ^ Bernays, Paul (1991). Axiomatic Set Theory. Dover Publications. ISBN 0-486-66637-9.

trichotomy, mathematics, trichotomy, states, that, every, real, number, either, positive, negative, zero, more, generally, binary, relation, trichotomous, exactly, holds, writing, this, stated, formal, logic, displaystyle, forall, forall, land, lnot, land, lno. In mathematics the law of trichotomy states that every real number is either positive negative or zero 1 More generally a binary relation 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 lt this is stated in formal logic as x X y X x lt y y lt x x y x lt y y lt x x y x lt y y lt x x y displaystyle forall x in X forall y in X x lt y land lnot y lt x land lnot x y lor lnot x lt y land y lt x land lnot x y lor lnot x lt y land lnot y lt x land x y Contents 1 Properties 2 Examples 3 Trichotomy on numbers 4 See also 5 ReferencesProperties EditA 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 this is a special case of a strict weak order 2 3 Examples EditOn the set X a b c the relation R a b a c b c is transitive and trichotomous and hence a strict total order On the same set the cyclic relation R a b b c c a is trichotomous but not transitive it is even antitransitive Trichotomy on numbers EditA 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 lt y y lt x or x y applies some authors even fix y to be zero 1 relying on the real number s additive linearly ordered group structure The latter is a group 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 integers and between rational numbers clarification needed The law does not hold in general in intuitionistic logic citation needed In Zermelo Fraenkel set theory and Bernays set theory the law of trichotomy holds between the cardinal numbers of well orderable sets even without the axiom of choice If the axiom of choice holds then trichotomy holds between arbitrary cardinal numbers because they are all well orderable in that case 4 See also EditBegriffsschrift contains an early formulation of the law of trichotomy Dichotomy Law of noncontradiction Law of excluded middle Three way comparisonReferences Edit a b Trichotomy Law at MathWorld Jerrold E Marsden amp Michael J Hoffman 1993 Elementary Classical Analysis page 27 W H Freeman and Company ISBN 0 7167 2105 8 H S Bear 1997 An Introduction to Mathematical Analysis page 11 Academic Press ISBN 0 12 083940 7 Bernays Paul 1991 Axiomatic Set Theory Dover Publications ISBN 0 486 66637 9 Retrieved from https en wikipedia org w index php title Law of trichotomy amp oldid 1096783955, wikipedia, wiki, book, books, library,

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