Logarithmic integral function: Difference between revisions
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{{Other uses|Distinction (disambiguation)}} | |||
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 [[equality (mathematics)|equal]]. In [[physics]] two things are distinct if they cannot be [[Topology|mapped]] to each other.<ref>{{Cite book|author=Martin, Keye|year=2010|chapter=Chapter 9: Domain Theory and Measurement: 9.6 Forms of Process Evolution|title=New Structures for Physics|series=Volume 813 of Lecture Notes in Physics|editor=Coecke, Bob|location=Heidelberg, Germany|publisher=Springer Verlag|pages=579–580|isbn=978-3-642-12820-2}}</ref> | |||
==Species or classes== | |||
{{Quote|[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<ref>{{Cite book|author=Locke, John|title = An Essay Concerning Human Understanding|chapter=Book 3: Chapter 6: Of the Names of Substances|url=http://www.ilt.columbia.edu/publications/Projects/digitexts/locke/understanding/chapter0306.html}}</ref>}} | |||
==In mathematics== | |||
===Example=== | |||
A [[quadratic equation]] over the [[complex number]]s has two [[root of a function|root]]s. | |||
The equation | |||
<math>x^{2} - 3x + 2 = 0</math> | |||
[[Factorization|factors]] as | |||
<math>(x - 1)(x - 2) = 0</math> | |||
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: | |||
<math>x^{2} - 2x + 1 = 0</math> | |||
factors as | |||
<math>(x - 1)(x - 1) = 0</math> | |||
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 [[mathematical proof|prove]] that two things ''x'' and ''y'' are distinct, it often helps to find some [[property (metaphysics)|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 number|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 (mathematics)|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 functional]]s. | |||
*In [[category theory]], if ''f'' is a [[functor]] between [[Category (mathematics)|categories]] '''C''' and '''D''', then ''f'' always maps [[morphism|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). | |||
<!--''I don't know what to say here, but there are issues. I can mention [[Leibniz's law]] (which we don't have an article on); this is the claim that distinct things can always be proved distinct as in the last section by some property. These ideas could be discussed under [[Identity (mathematics)|Identity]]. There is some discussion at [[Identity and change]].'' | |||
--> | |||
==See also== | |||
{{Wiktionary|distinct}} | |||
==Notes== | |||
{{Reflist}} | |||
[[Category:Elementary mathematics]] | |||
Revision as of 21:55, 20 December 2013
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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. In physics two things are distinct if they cannot be mapped to each other.[1]
Species or classes
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In mathematics
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).
See also
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Notes
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