Distinct 47285 224102023 2008-07-07T09:13:38Z Mhaitham.shammaa 1608031 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]]. ===Example=== A [[quadratic equation]] over the [[complex number]]s always has two [[Root (mathematics)|root]]s. The equation : ''x''<sup>2</sup> &minus; 3''x'' + 2 = 0 [[Factorization|factors]] as : (''x'' &minus; 1)(''x'' &minus; 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''<sup>2</sup> &minus; 2''x'' + 1 = 0 factors as : (''x'' &minus; 1)(''x'' &minus; 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 [[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 (up to isomorphism). <!--''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]].'' --> [[Category:Elementary mathematics]] ==See also== {{Wiktionarypar|distinct}} *[[distinction]] *[[list of distinctions]] [[ar:متمايز]] [[de:Paarweise verschieden]] [[th:ภาวะแตกต่าง]]