World Library  
Flag as Inappropriate
Email this Article


Article Id: WHEBN0000047285
Reproduction Date:

Title: Distinct  
Author: World Heritage Encyclopedia
Language: English
Subject: Distinction, Different, Separable polynomial, Religious belief, Injective function
Publisher: World Heritage Encyclopedia


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. In physics two things are distinct if they cannot be mapped to each other.[1]


  • Species or classes 1
  • In mathematics 2
    • Example 2.1
    • Proving distinctness 2.2
  • See also 3
  • Notes 4

Species or classes

[I]t is plain that our distinct species are nothing but distinct complex ideas, with distinct names annexed to them. It is true every substance that exists has its peculiar constitution, whereon depend those sensible qualities and powers we observe in it; but the ranking of things into species (which is nothing but sorting them under several titles) is done by us according to the ideas that we have of them: which, though sufficient to distinguish them by names, so that we may be able to discourse of them when we have them not present before us; yet if we suppose it to be done by their real internal constitutions, and that things existing are distinguished by nature into species, by real essences, according as we distinguish them into species by names, we shall be liable to great mistakes.
—John Locke, An Essay Concerning Human Understanding[2]

In mathematics


A quadratic equation over the complex numbers has two roots.

The equation

x^{2} - 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:

x^{2} - 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

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,

See also


  1. ^ 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.  
  2. ^ Locke, John. "Book 3: Chapter 6: Of the Names of Substances". An Essay Concerning Human Understanding. 
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.

Copyright © World Library Foundation. All rights reserved. eBooks from Project Gutenberg are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.