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

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A quadratic equation over the complex numbers always 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

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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 (up to isomorphism).

Elementary mathematics

See also

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distinction