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

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

1. The solution set of the single equation ${\displaystyle f(x)=x}${\displaystyle f(x)=x} is the set  {0}.

2. For any non-zero polynomial ${\displaystyle f}${\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.

In algebraic geometry, solution sets are used to define the Zariski topology. See affine varieties.

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 ${\displaystyle {(x_{j})}_{j\in J}}${\displaystyle {(x_{j})}_{j\in J}}, supposed to take values in respective spaces ${\displaystyle {(X_{j})}_{j\in J}}${\displaystyle {(X_{j})}_{j\in J}}, is the set S of all solutions to the relations E, where a solution ${\displaystyle x^{(k)}}${\displaystyle x^{(k)}} is a family of values ${\displaystyle {(x_{j}^{(k)})}_{j\in J}\in \prod _{j\in J}X_{j}}${\displaystyle {(x_{j}^{(k)})}_{j\in J}\in \prod _{j\in J}X_{j}} such that substituting ${\displaystyle {(x_{j})}_{j\in J}}${\displaystyle {(x_{j})}_{j\in J}} by ${\displaystyle x^{(k)}}${\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 f_i if interpreted as the set of equations f_i(x)=0.

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

• equation solving

• Equations

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