- Solution set
In
mathematics , a solution set is a set of possible values that a variable can take on in order to satisfy a given set of conditions (which may include equations and inequalities).Formally, for a collection of
polynomial s f_i} over some ring R, a "solution set" is defined to be the set xin R:forall iin I, f_i(x)=0}.Examples
1. The solution set of f(x):=x over the real numbers is the set {0}.
2. For any non-zero polynomial f over the
complex numbers in one variable, the solution set is made up of finitely many points. However, for a complex polynomial in more than one variable the solution set has no isolated points.In non mathematical terms, it refers to the web development and branding company [http://www.SolutionSet.com SolutionSet] based in Palo Alto, CA
Remarks
In
algebraic geometry solution sets are used to define theZariski topology . Seeaffine varieties .Other meanings
More generally, the solution set to an arbitrary collection "E" of relations ("Ei") ("i" varying in some index set "I") for a collection of unknowns x_j)}_{jin J}, supposed to take values in respective spaces X_j)}_{jin J}, is the set "S" of all solutions to the relations "E", where a solution x^{(k)} is a family of values x^{(k)}_j)}_{jin J}in prod_{jin J} X_j such that substituting x_j)}_{jin J} by x^{(k)} in the collection "E" makes all relations "true".
(Instead of relations depending on unknowns, one should speak more correctly of predicates, the collection "E" is their
logical conjunction , and the solution set is theinverse image of the boolean value "true" by the associatedboolean-valued function .)The above meaning is a special case of this one, if the set of polynomials "fi" if interpreted as the set of equations "fi(x)=0".
Examples
* The solution set for " E = { x+y = 0 } " w.r.t. x,y)inmathbb R^2 is " S = { (a,-a) ; a ∈ R } ".
* The solution set for " E = { x+y = 0 } " w.r.t. xinmathbb R is " S = { -y } ". (Here, "y" is not "declared" as an unknown, and thus to be seen as aparameter on which the equation, and therefore the solution set, depends.)
* The solution set for E = { sqrt x le 4 } w.r.t. xinmathbb R is the interval " S = [0,2] " (since the equation (inequality) is not well defined for negative numbers).
* The solution set for E = { exp(i x) = 1 } w.r.t. xinmathbb C is " S" = 2 π Z (seeEuler's identity ).other example:6z+5>35ee also
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equation solving
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