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. New Structures for Physics. Volume 813 of Lecture Notes in Physics. Heidelberg, Germany: Springer Verlag. pp. 579–580. ISBN 978-3-642-12820-2.