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 polynomials $\backslash \{f\_i\backslash \}$ over some ring $R$, a solution set is defined to be the set $\backslash \{x\backslash in\; R:\backslash forall\; i\backslash in\; I,\; f\_i(x)=0\backslash \}$.

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.

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 relations (E_i) (i varying in some index set I) for a collection of unknowns $\{(x\_j)\}\_\{j\backslash in\; J\}$, supposed to take values in respective spaces $\{(X\_j)\}\_\{j\backslash in\; J\}$, is the set S of all solutions to the relations E, where a solution $x^\{(k)\}$ is a family of values $\{(x^\{(k)\}\_j)\}\_\{j\backslash in\; J\}\backslash in\; \backslash prod\_\{j\backslash in\; J\}\; X\_j$ such that substituting $\{(x\_j)\}\_\{j\backslash in\; J\}$ 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 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_i if interpreted as the set of equations f_i(x)=0.

Examples

• The solution set for E = { x+y = 0 } w.r.t. $(x,y)\backslash in\backslash mathbb\; R^2$ is S = { (a,-a) ; a ∈ R } .
• The solution set for E = { x+y = 0 } w.r.t. $x\backslash in\backslash mathbb\; R$ 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 $E\; =\; \backslash \{\; \backslash sqrt\; x\; \backslash le\; 4\; \backslash \}$ w.r.t. $x\backslash in\backslash mathbb\; R$ is the interval S = ^* (since the equation (inequality) is not well defined for negative numbers).
• The solution set for $E\; =\; \backslash \{\; \backslash exp(i\; x)\; =\; 1\; \backslash \}$ w.r.t. $x\backslash in\backslash mathbb\; C$ is S = 2 π Z (see Euler's identity).

See also

• equation solving

Equations

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