Gaussian integral: Difference between revisions

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Re-introduced the Gaussian graph SVG. <Investigation: Oct 2012: Anonymous user changed the code making it unusable; Following Edit: A user deletes the code entirely.
mismatch between definite and indefinite integrals
 
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In [[mathematics]], a '''solution set''' is the [[set (mathematics)|set]] of values that satisfy a given set of equations or inequalities.
 
For example, for a set <math>\{f_i\}</math> of [[polynomial]]s over a [[Ring (mathematics)|ring]] <math>R</math>,
the solution set is the subset of <math>R</math> on which the polynomials all vanish (evaluate to 0), formally
:<math>\{x\in R:\forall i\in I, f_i(x)=0\}.\ </math>
 
== Examples ==
1. The solution set of the single equation <math>f(x)=x</math> is the set&nbsp; {0}.
 
2. For any non-zero polynomial <math>f</math> over the [[complex numbers]] in one variable, the solution set is made up of finitely many points.
 
3. However, for a complex polynomial in more than one variable the solution set has no isolated points.
 
== Remarks ==
In [[algebraic geometry]], solution sets are used to define the [[Zariski topology]]. See [[affine varieties]].
 
== Other meanings ==
More generally, the '''solution set''' to an arbitrary collection ''E'' of [[relation (mathematics)|relation]]s (''E<sub>i</sub>'') (''i'' varying in some index set ''I'') for a collection of unknowns <math>{(x_j)}_{j\in J}</math>, supposed to take values in respective spaces <math>{(X_j)}_{j\in J}</math>, is the set ''S'' of all solutions to the relations ''E'', where a solution <math>x^{(k)}</math> is a family of values <math>{(x^{(k)}_j)}_{j\in J}\in \prod_{j\in J} X_j</math> such that substituting <math>{(x_j)}_{j\in J}</math> by <math>x^{(k)}</math> in the collection ''E'' makes all relations "true".
 
(Instead of relations depending on unknowns, one should speak more correctly of [[Predicate (mathematics)|predicate]]s, the collection ''E'' is their [[logical conjunction]], and the solution set is the [[inverse image]] of the boolean value ''true'' by the associated [[boolean-valued function]].)
 
The above meaning is a special case of this one, if the set of polynomials ''f<sub>i</sub>'' if interpreted as the set of equations ''f<sub>i</sub>(x)=0''.
 
===Examples===
 
* The solution set for '' E = { x+y = 0 } '' w.r.t. <math>(x,y)\in\mathbb R^2</math> is '' S = { (a,-a) ; a ∈ '''R''' } ''.
* The solution set for '' E = { x+y = 0 } '' w.r.t. <math>x\in\mathbb R</math> is '' S = { -y } ''. (Here, ''y'' is not "declared" as an unknown, and thus to be seen as a [[parameter]] on which the equation, and therefore the solution set, depends.)
* The solution set for <math> E = \{ \sqrt x \le 4 \} </math> w.r.t. <math>x\in\mathbb R</math> is the interval '' S = [0,2] '' (since <math>\sqrt x</math> is undefined for negative values of ''x'').
* The solution set for <math> E = \{ \exp(i x) = 1 \} </math> w.r.t. <math>x\in\mathbb C</math> is '' S'' = 2 π '''Z''' (see [[Euler's identity]]).
 
==See also==
* [[equation solving]]
 
{{DEFAULTSORT:Solution Set}}
[[Category:Equations]]

Latest revision as of 15:39, 11 January 2015

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The Vintage, Vital and Extremely Shorter Boots occur in sand, chocolate and black.

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