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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.## 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.## See also

The equation

- x
^{2}− 3x + 2 = 0

- (x − 1)(x − 2) = 0

In contrast, the equation:

- x
^{2}− 2x + 1 = 0

- (x − 1)(x − 1) = 0

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.)

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 (up to isomorphism).

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Last updated on Wednesday October 08, 2008 at 14:31:37 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Wednesday October 08, 2008 at 14:31:37 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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