 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
x^{2} − 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:
x^{2} − 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,
 The Hahn–Banach theorem says (among other things) that distinct elements of a Banach space can be proved to be distinct using only linear functionals.
 In category theory, if f is a functor between categories C and D, then f always maps isomorphic objects to isomorphic objects. Thus, one way to show two objects of C are distinct (up to isomorphism) is to show that their images under f are distinct (i.e. not isomorphic).
See also
 Distinction
 List of distinctions
Categories:
Wikimedia Foundation. 2010.
Look at other dictionaries:
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