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Solution set

## Summary

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

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

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

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

## Examples

1. The solution set of the single equation ${\displaystyle x=0}$  is the set {0}.
2. For any non-zero polynomial ${\displaystyle f}$  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 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 (Ei) (i varying in some index set I) for a collection of unknowns ${\displaystyle {(x_{j})}_{j\in J}}$ , supposed to take values in respective spaces ${\displaystyle {(X_{j})}_{j\in J}}$ , is the set S of all solutions to the relations E, where a solution ${\displaystyle x^{(k)}}$  is a family of values ${\textstyle {\left(x_{j}^{(k)}\right)}_{j\in J}\in \prod _{j\in J}X_{j}}$  such that substituting ${\displaystyle {\left(x_{j}\right)}_{j\in J}}$  by ${\displaystyle x^{(k)}}$  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 fi if interpreted as the set of equations fi(x)=0.

### Examples

• The solution set for E = { x+y = 0 } with respect to ${\displaystyle (x,y)\in \mathbb {R} ^{2}}$  is S = { (a,−a) : aR }.
• The solution set for E = { x+y = 0 } with respect to ${\displaystyle x\in \mathbb {R} }$  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 ${\displaystyle E=\{{\sqrt {x}}\leq 4\}}$  with respect to ${\displaystyle x\in \mathbb {R} }$  is the interval S = [0,2] (since ${\displaystyle {\sqrt {x}}}$  is undefined for negative values of x).
• The solution set for ${\displaystyle E=\{e^{ix}=1\}}$  with respect to ${\displaystyle x\in \mathbb {C} }$  is S = 2πZ (see Euler's identity).