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Inmathematics, thesolution set of asystem of equations orinequality is theset of all its solutions, that is the values that satisfy all equations and inequalities.^[1] Also, the solution set or thetruth set of a statement or apredicate is the set of all values that satisfy it.

If there is no solution, the solution set is theempty set.^[2]

• The solution set of the single equation${\displaystyle x=0}$ is thesingleton set${\displaystyle \{0\}}$.
• Since there do not exist numbers${\displaystyle x}$ and${\displaystyle y}$ making the two equations simultaneously true, the solution set of this system is theempty set${\displaystyle \mathrm{\varnothing }}$.
• The solution set of aconstrained optimization problem is itsfeasible region.
• The truth set of the predicate${\displaystyle P(n):n\mathrm{i}\mathrm{s}\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{n}}$ is${\displaystyle \{2,4,6,8,\dots \}}$.

Inalgebraic geometry, solution sets are calledalgebraic sets if there are no inequalities. Over thereals, and with inequalities, there are calledsemialgebraic sets.

More generally, thesolution set to an arbitrary collectionE ofrelations (E_i) (i varying in some index setI) 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 setS of all solutions to the relationsE, where a solution${\displaystyle x^{(k)}}$ is a family of values${\left(x_j^{(k)}\right)}_{j\in J}\in \prod _{j\in J}{X}_j$ such that substituting${\displaystyle {\left(x_j\right)}_{j\in J}}$ by${\displaystyle x^{(k)}}$ in the collectionE makes all relations "true".

(Instead of relations depending on unknowns, one should speak more correctly ofpredicates, the collectionE is theirlogical conjunction, and the solution set is theinverse image of the Boolean valuetrue by the associatedBoolean-valued function.)

The above meaning is a special case of this one, if the set of polynomialsf_i if interpreted as the set of equationsf_i(x)=0.

• The solution set forE = {x+y = 0 } with respect to${\displaystyle (x,y)\in {\mathbb{R}}^2}$ isS = { (a,−a) :a ∈R }.
• The solution set forE = {x+y = 0 } with respect to${\displaystyle x\in \mathbb{R}}$ isS = { −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${\displaystyle E=\{\sqrt{x}\le 4\}}$ with respect to${\displaystyle x\in \mathbb{R}}$ is the intervalS = [0,16] (since${\displaystyle \sqrt{x}}$ is undefined for negative values ofx).
• The solution set for${\displaystyle E=\{e^{ix}=1\}}$ with respect to${\displaystyle x\in \mathbb{C}}$ isS = 2πZ (seeEuler's identity).

• Equation solving
• Extraneous and missing solutions
• Equaliser (mathematics)

• Equations

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