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

inequalities Inequality may refer to: Economics * Attention inequality, unequal distribution of attention across users, groups of people, issues in etc. in attention economy * Economic inequality, difference in economic well-being between population groups * ...

. For example, for a set of

polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An ex ...

s over a

ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...

, the solution set is the subset of on which the polynomials all vanish (evaluate to 0), formally : The

feasible region In mathematical optimization, a feasible region, feasible set, search space, or solution space is the set of all possible points (sets of values of the choice variables) of an optimization problem that satisfy the problem's constraints, potent ...

of a

constrained optimization In mathematical optimization, constrained optimization (in some contexts called constraint optimization) is the process of optimizing an objective function with respect to some variables in the presence of constraints on those variables. The obj ...

problem is the solution set of the constraints.

Examples

# The solution set of the single equation

x=0

is the set . # For any non-zero polynomial

f

over the

complex numbers In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...

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 set Algebraic may refer to any subject related to algebra in mathematics and related branches like algebraic number theory and algebraic topology. The word algebra itself has several meanings. Algebraic may also refer to: * Algebraic data type, a dat ...

s if there are no inequalities. Over the reals, and with inequalities, there are called

semialgebraic set In mathematics, a semialgebraic set is a subset ''S'' of ''Rn'' for some real closed field ''R'' (for example ''R'' could be the field of real numbers) defined by a finite sequence of polynomial equations (of the form P(x_1,...,x_n) = 0) and ineq ...

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

_

, supposed to take values in respective spaces

_

, is the set ''S'' of all solutions to the relations ''E'', where a solution

x^

is a family of values

_\in \prod_ X_j

such that substituting

_

x^

in the collection ''E'' makes all relations "true". (Instead of relations depending on unknowns, one should speak more correctly of

predicate Predicate or predication may refer to: * Predicate (grammar), in linguistics * Predication (philosophy) * several closely related uses in mathematics and formal logic: **Predicate (mathematical logic) **Propositional function **Finitary relation, ...

s, the collection ''E'' is their

logical conjunction In logic, mathematics and linguistics, And (\wedge) is the truth-functional operator of logical conjunction; the ''and'' of a set of operands is true if and only if ''all'' of its operands are true. The logical connective that represents thi ...

, and the solution set is the

inverse image In mathematics, the image of a function is the set of all output values it may produce. More generally, evaluating a given function f at each element of a given subset A of its domain produces a set, called the "image of A under (or through ...

of the boolean value ''true'' by the associated

boolean-valued function A Boolean-valued function (sometimes called a Predicate (logic), predicate or a proposition) is a function (mathematics), function of the type f : X → B, where X is an arbitrary Set (mathematics), set and where B is a Boolean domain, i.e. a gene ...

.) 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'' = with respect to

(x,y)\in \R^2

is ''S'' = . * The solution set for ''E'' = with respect to

x \in \R

is ''S'' = . (Here, ''y'' is not "declared" as an unknown, and thus to be seen as a

parameter A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...

on which the equation, and therefore the solution set, depends.) * The solution set for

E = \

with respect to

x\in\R

is the interval ''S'' = ,2(since

\sqrt x

is undefined for negative values of ''x''). * The solution set for

E = \

with respect to

x\in\Complex

is ''S'' = 2πZ (see Euler's identity).

Examples

Remarks

Other meanings

Examples

See also