|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| {{Refimprove|date=October 2013}}
| | I am 39 years old and my name is Armand Falbo. I life in Vlaardingen (Netherlands).<br><br>my blog post :: [http://home.genplace.com/link.php?url=http://mypinboardrecipes.com/ best Recipes best weather app for android] |
| {{wiktionary|trivial}}
| |
| In [[mathematics]], the adjective '''''trivial''''' is frequently used for [[Category theory|objects]] (for example, [[group (mathematics)|group]]s or [[topological space]]s) that have a very simple structure. The noun '''''triviality''''' usually refers to a simple technical aspect of some proof or definition. The origin of the term in mathematical language comes from the medieval [[Trivium (education)|trivium]] curriculum. The antonym ''nontrivial'' is commonly used by engineers and mathematicians to indicate a statement or theorem that is not obvious or easy to prove.
| |
| | |
| ==Trivial and nontrivial solutions==
| |
| In mathematics, the term trivial is frequently used for objects (for examples, groups or topological spaces) that have a very simple structure. For non-mathematicians, they are sometimes more difficult to visualize or understand than other, more complicated objects{{citation needed|date=June 2013}}.
| |
| | |
| Examples include:
| |
| *[[empty set]]: the [[Set (mathematics)|set]] containing no members
| |
| *[[trivial group]]: the mathematical [[group (mathematics)|group]] containing only the [[identity element]]
| |
| *[[trivial ring]]: a [[ring (mathematics)|ring]] defined on a [[singleton set]].
| |
| ''Trivial'' also refers to solutions to an [[equation]] that has a very simple structure, but for the sake of completeness cannot be omitted. These solutions are called the '''trivial solution'''. For example, consider the [[differential equation]]
| |
| :<math>y'=y</math>
| |
| where ''y'' = ''f''(''x'') is a [[function (mathematics)|function]] whose [[derivative]] is ''y''′. The trivial solution is
| |
| :''y'' = 0, the [[0 (number)#Related mathematical terms|zero function]] | |
| while a '''nontrivial''' solution is
| |
| :''y'' (''x'') = e<sup>''x''</sup>, the [[exponential function]]. | |
| | |
| The differential equation <math>f''(x)=-\lambda f(x)</math> with boundary conditions <math>f(0) = f(L) = 0</math> is important in math and physics, for example describing a [[particle in a box]] in quantum mechanics, or [[standing wave]]s on a string. It always has the solution <math>f(x) = 0</math>. This solution is considered obvious and is called the "trivial" solution. In some cases, there may be other solutions ([[Sine wave|sinusoid]]s), which are called "nontrivial".<ref>[http://books.google.com/books?id=YRT0W31HbTwC&pg=PA309 Introduction to partial differential equations with applications, by Zachmanoglou and Thoe, p309]</ref>
| |
| | |
| Similarly, mathematicians often describe [[Fermat's Last Theorem]] as asserting that there are no nontrivial integer solutions to the equation <math>a^n + b^n = c^n</math> when ''n'' is greater than 2. Clearly, there ''are'' some solutions to the equation. For example, <math>a=b=c=0</math> is a solution for any ''n'', but such solutions are all obvious and uninteresting, and hence "trivial".
| |
| | |
| ==Triviality in mathematical reasoning==
| |
| ''Trivial'' may also refer to any easy [[Proof by exhaustion|case]] of a proof, which for the sake of completeness cannot be ignored. For instance, proofs by [[mathematical induction]] have two parts: the "base case" that shows that the theorem is true for a particular initial value such as ''n'' = 0 or ''n'' = 1 and then an inductive step that shows that if the theorem is true for a certain value of ''n'', it is also true for the value ''n'' + 1. The base case is often trivial and is identified as such, although there are cases where the base case is difficult but the inductive step is trivial. Similarly, one might want to prove that some property is possessed by all the members of a certain set. The main part of the proof will consider the case of a nonempty set, and examine the members in detail; in the case where the set is empty, the property is trivially possessed by all the members, since there are none. (See also [[Vacuous truth]].)
| |
| | |
| A common joke in the mathematical community is to say that "trivial" is synonymous with "proved" — that is, any theorem can be considered "trivial" once it is known to be true. Another joke concerns two mathematicians who are discussing a theorem; the first mathematician says that the theorem is "trivial". In response to the other's request for an explanation, he then proceeds with twenty minutes of exposition. At the end of the explanation, the second mathematician agrees that the theorem is trivial. These jokes point out the subjectivity of judgments about triviality. The joke also applies when the first mathematician says the theorem is trivial, but is unable to prove it himself. Often, as a joke, the theorem is then referred to as "intuitively obvious." Someone experienced in [[calculus]], for example, would consider the statement that
| |
| :<math>\int_0^1 x^2\, dx = \frac{1}{3}</math>
| |
| to be trivial. To a beginning student of calculus, though, this may not be obvious at all.
| |
| | |
| Triviality also depends on context. A proof in [[functional analysis]] would probably, given a number, trivially assume the existence of a larger number. When proving basic results about the natural numbers in [[elementary number theory]] though, the proof may very well hinge on the remark that any natural number has a successor (which should then in itself be proved or taken as an [[axiom]], see [[Peano's axioms]]).
| |
| | |
| === Trivial proofs ===
| |
| In some texts, a ''trivial proof'' refers to a statement involving a [[Material implication (rule of inference)|material implication]] where the [[consequent]], or ''Q'', in ''P''→''Q'', is always true.<ref name="Chartrand">{{cite book|last=Zhang|first=Gary Chartrand, Albert D. Polimeni, Ping|title=Mathematical proofs : a transition to advanced mathematics|year=2008|publisher=Pearson/Addison Wesley|location=Boston|isbn=978-0-3-2139053-0|page=68|edition=2nd ed.}}</ref> Here, the proof follows simply from noting that ''Q'' is always true, as the implication is then true regardless of the truth value of the [[antecedent (logic)|antecedent]], ''P''.<ref name="Chartrand" />
| |
| | |
| A related concept is a [[vacuous truth|vacuous proof]], where the antecedent, ''P'', in the material implication ''P''→''Q'' is always false.<ref name="Chartrand" /> Here, the implication is always true regardless of the truth value of the consequent, ''Q''.<ref name="Chartrand" />
| |
| | |
| == Examples ==
| |
| *In [[mathematics]], it is often important to find [[divisor|factors]] of an integer number N. Any number N has four obvious factors: ±1 and ±N. These are called "trivial factors". Any other factor, if any exist, would be called "nontrivial".<ref>[http://books.google.com/books?id=lIvPz7k41SEC&pg=PA250 Number theory for computing, by Song Y. Yan, p250]</ref>
| |
| | |
| *The [[matrix (mathematics)|matrix]] equation AX=0, where A is a fixed matrix, X is an unknown vector, and 0 is the zero vector, has an obvious solution X=0. This is called the "trivial solution". If it has other solutions X≠0, they would be called "nontrivial"<ref>[http://books.google.com/books?id=KWAqz3m2xYUC&pg=PA502 Mathematics for engineers and scientists, by Alan Jeffrey, p502]</ref>
| |
| | |
| *In the mathematics of [[group theory]], there is a very simple group with just one element in it; this is often called the "trivial group". All other groups, which are more complicated, are called "nontrivial".
| |
| | |
| *In the [[graph theory]] the trivial graph is a graph which has only 1 vertex and no edges.
| |
| | |
| *[[Database theory]] has a concept called [[functional dependency]], written <math> X \to Y </math>. It is obvious that the dependence <math> X \to Y </math> is true if ''Y'' is a [[subset]] of ''X'', so this type of dependence is called "trivial". All other dependences, which are less obvious, are called "nontrivial".
| |
| | |
| == See also ==
| |
| * [[Initial and terminal objects]]
| |
| * [[Degeneracy (mathematics)|Degeneracy]]
| |
| * [[Pathological (mathematics)|Pathological]]
| |
| | |
| ==References==
| |
| {{Reflist}}
| |
| | |
| ==External links==
| |
| *[http://mathworld.wolfram.com/Trivial.html Trivial entry at MathWorld]
| |
| | |
| {{DEFAULTSORT:Trivial (Mathematics)}}
| |
| [[Category:Mathematical terminology]]
| |