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Learn more about "Distinct"
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DistinctTwo 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 equality (mathematics)|equal.
Example
A quadratic equation over the complex numbers sometimes has two Root (mathematics)|roots.
The equation
- ''x''2 − 3''x'' + 2 = 0
Factorization|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 − 2''x'' + 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 mathematical proof|prove that two things ''x'' and ''y'' are distinct, it often helps to find some property (metaphysics)|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 number|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 (mathematics)|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 Category (mathematics)|categories '''C''' and '''D''', then ''f'' always maps morphism|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 (up to isomorphism).
Category:Elementary mathematics
See also
- distinction
- list of distinctions
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