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Encyclopedia

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,

See also

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Wiktionary

Up to date as of January 15, 2010

Definition from Wiktionary, a free dictionary

Contents

English

Etymology

< Middle English < Old French < Latin distinctus, pp. of distinguere (to distinguish); see distinguish.

Pronunciation

Adjective

distinct (comparative more distinct, superlative most distinct)

Positive
distinct

Comparative
more distinct

Superlative
most distinct

  1. Very clear.
    Her voice was distinct despite the heavy traffic.
  2. Different from one another.
    The letters in the word "clear" are distinct, while the letters in the word "distinct" are not.
  3. Noticeably different.
    Olga's voice is quite distinct because of her accent.

Synonyms

Antonyms

Related terms

Translations

The translations below need to be checked and inserted above into the appropriate translation tables, removing any numbers. Numbers do not necessarily match those in definitions. See instructions at Help:How to check translations.

External links

  • distinct in Webster’s Revised Unabridged Dictionary, G. & C. Merriam, 1913
  • distinct in The Century Dictionary, The Century Co., New York, 1911

French

Pronunciation

Adjective

distinct m. (f. distincte, m. plural distincts, f. plural distinctes)

  1. distinct

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