Solution set
# Solution set

In mathematics, a solution set is the set of values that satisfy a given set of equations or inequalities.

For example, for a set $\{f_{i}\}$ of polynomials over a ring $R$, the solution set is the subset of $R$ on which the polynomials all vanish (evaluate to 0), formally

    $\{x\in R:\forall i\in I,f_{i}(x)=0\}$

The feasible region of a constrained optimization problem is the solution set of the constraints.

## Examples

* The solution set of the single equation $x=0$ is the set {0}.
* 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 called algebraic sets if there are no inequalities. Over the reals, and with inequalities, there are called semialgebraic sets.

## 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 ${(x_{j})}_{j\in J}$, supposed to take values in respective spaces ${(X_{j})}_{j\in J}$, is the set S of all solutions to the relations E, where a solution $x^{(k)}$ is a family of values $(x_j^{(k)})_{j∈J}∈∏_{j∈J}X_j{\textstyle$ {\left(x_{j}^{(k)}\right)}_{j\in J}\in \prod _{j\in J}X_{j}} such that substituting ${\left(x_{j}\right)}_{j\in J}$ 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 } with respect to $(x,y)\in \mathbb {R} ^{2}$ is S = { (a,−a) : a ∈ R }.
* The solution set for E = { x+y = 0 } with respect to $x\in \mathbb {R}$ 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 $E=\{{\sqrt {x}}\leq 4\}$ with respect to $x\in \mathbb {R}$ is the interval S = [0,2] (since ${\sqrt {x}}$ is undefined for negative values of x).
* The solution set for $E=\{e^{ix}=1\}$ with respect to $x\in \mathbb {C}$ is S = 2πZ (see Euler's identity).
