Distinct


Distinct

Two or more things are distinct if no two of them are the same thing. In mathematics, two things are called distinct if they are not equal.

Example

A quadratic equation over the complex numbers sometimes has two roots.

The equation

x2 − 3x + 2 = 0

factors as

(x − 1)(x − 2) = 0

and thus has as roots x = 1 and x = 2. Since 1 and 2 are not equal, these roots are distinct.

In contrast, the equation:

x2 − 2x + 1 = 0

factors as

(x − 1)(x − 1) = 0

and thus has as roots x = 1 and x = 1. Since 1 and 1 are (of course) equal, the roots are not distinct; they coincide.

In other words, the first equation has distinct roots, while the second does not. (In the general theory, the discriminant is introduced to explain this.)

Proving distinctness

In order to prove that two things x and y are distinct, it often helps to find some property that one has but not the other. For a simple example, if for some reason we had any doubt that the roots 1 and 2 in the above example were distinct, then we might prove this by noting that 1 is an odd number while 2 is even. This would prove that 1 and 2 are distinct.

Along the same lines, one can prove that x and y are distinct by finding some function f and proving that f(x) and f(y) are distinct. This may seem like a simple idea, and it is, but many deep results in mathematics concern when you can prove distinctness by particular methods. For example,

See also


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  • distinct — 1 Distinct, separate, several, discrete are comparable when used in reference to two or more things (sometimes persons) and in the sense of not being individually the same. Distinct always implies a capacity for being distinguished by the eye or… …   New Dictionary of Synonyms

  • distinct — distinct, incte [ distɛ̃(kt), ɛ̃kt ] adj. • 1308; lat. distinctus, de distinguere 1 ♦ Qui ne se confond pas avec qqch. d analogue, de voisin. ⇒ autre, différent, indépendant, séparé. Problèmes, domaines distincts. « la politique n est pas… …   Encyclopédie Universelle

  • distinct — distinct, distinctive 1. Both words are related to the verb distinguish, but distinct means essentially ‘separate, different’ (The word has several distinct meanings) or ‘unmistakable, decided’ (She has a distinct impression of being watched),… …   Modern English usage

  • distinct — DISTÍNCT, Ă, distincţi, te, adj. 1. Care se deosebeşte prin anumite trăsături proprii de alte lucruri de acelaşi fel sau asemănătoare; deosebit, diferit. 2. (Adesea adverbial) Clar, evident, lămurit, desluşit. – Din fr. distinct, lat. distinctus …   Dicționar Român

  • Distinct — Dis*tinct , a. [L. distinctus, p. p. of distinguere: cf. F. distinct. See {Distinguish}.] 1. Distinguished; having the difference marked; separated by a visible sign; marked out; specified. [Obs.] [1913 Webster] Wherever thus created for no place …   The Collaborative International Dictionary of English

  • distinct — I (clear) adjective apparent, clarus, clear cut, clear to the mind, clear to the senses, clearly defined, concrete, conspicuous, crystal clear, definite, distinctus, distinguishable, easily perceived, easily understood, eidetic, evident, explicit …   Law dictionary

  • distinct — distinct, te (di stin, stin kt ; il y a trois manières différentes de prononcer ce mot au masculin : les uns disent di stinkt , les autres di stink ; d autres enfin di stin ; cette dernière manière a pour elle l analogie ; c était celle du temps… …   Dictionnaire de la Langue Française d'Émile Littré

  • distinct — [di stiŋkt′] adj. [ME & OFr < L distinctus, pp. of distinguere: see DISTINGUISH] 1. not alike; different 2. not the same; separate; individual 3. clearly perceived or marked off; clear; plain [a distinct image] 4. well defined; unmistakable;… …   English World dictionary

  • distinct — DISTINCT, INCTE. adject. Différent, séparé d un autre. Ce sont deux choses bien distinctes. Il faut que les articles d un compte soient bien distincts. [b]f♛/b] Il signifie aussi, Clair et net. Un son distinct, une voix distincte. Une vue… …   Dictionnaire de l'Académie Française 1798

  • distinct — [adj1] apparent, obvious audible, categorical, clean cut, clear, clear cut, decided, definite, enunciated, evident, explicit, express, incisive, lucid, manifest, marked, noticeable, palatable, patent, perspicuous, plain, prescribed, recognizable …   New thesaurus

  • distinct — Distinct, [distin]cte. adj. v. Different, separé d un autre. Ce sont deux choses bien distinctes. il faut que les articles d un compte soient distincts. Il signifie aussi, Clair & net. Un son distinct. une voix distincte. une veüe distincte. en… …   Dictionnaire de l'Académie française


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