# Distinct (mathematics)

In mathematics, two things are called **distinct** if they are not equal. In physics two things are distinct if they cannot be mapped to each other.[1]

## Example

A quadratic equation over the complex numbers has two roots.

The equation

factors as

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:

factors as

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).

## Notes

- Martin, Keye (2010). "Chapter 9: Domain Theory and Measurement: 9.6 Forms of Process Evolution". In Coecke, Bob (ed.).
*New Structures for Physics*. Volume 813 of Lecture Notes in Physics. Heidelberg, Germany: Springer Verlag. pp. 579–580. ISBN 978-3-642-12820-2.