Solution set
Encyclopedia
In mathematics
, a solution set is the set of values that satisfy a given set of equations or inequalities.
For example, for a set
of polynomial
s over a ring
,
the solution set is the subset of
on which the polynomials all vanish (evaluate to 0), formally
also 5<10 is 5 and any between 10
is the set {0}.
2. For any non-zero polynomial
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.
, solution sets are used to define the Zariski topology
. See affine varieties.
s (Ei) (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 
is a family of values 
such that substituting 
by 
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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a solution set is the set of values that satisfy a given set of equations or inequalities.
For example, for a set
of polynomialPolynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
s over a ring
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
,the solution set is the subset of
on which the polynomials all vanish (evaluate to 0), formallyalso 5<10 is 5 and any between 10Examples
1. The solution set of the single equation is the set {0}.2. For any non-zero polynomial
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 geometryAlgebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
, solution sets are used to define the Zariski topology
Zariski topology
In algebraic geometry, the Zariski topology is a particular topology chosen for algebraic varieties that reflects the algebraic nature of their definition. It is due to Oscar Zariski and took a place of particular importance in the field around 1950...
. See affine varieties.
Other meanings
More generally, the solution set to an arbitrary collection E of relationRelation (mathematics)
In set theory and logic, a relation is a property that assigns truth values to k-tuples of individuals. Typically, the property describes a possible connection between the components of a k-tuple...
s (Ei) (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 is a family of values such that substituting by 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
Logical conjunction
In logic and mathematics, a two-place logical operator and, also known as logical conjunction, results in true if both of its operands are true, otherwise the value of false....
, and the solution set is the inverse image of the boolean value true by the associated boolean-valued function
Boolean-valued function
A boolean-valued function, in some usages is a predicate or a proposition, is a function of the type f : X → B, where X is an arbitrary set and where B is a boolean domain....
.)
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 } w.r.t.
is S = { (a,-a) ; a ∈ R } . - The solution set for E = { x+y = 0 } w.r.t.
is S = { -y } . (Here, y is not "declared" as an unknown, and thus to be seen as a parameterParameterParameter from Ancient Greek παρά also “para” meaning “beside, subsidiary” and μέτρον also “metron” meaning “measure”, can be interpreted in mathematics, logic, linguistics, environmental science and other disciplines....
on which the equation, and therefore the solution set, depends.) - The solution set for
w.r.t. 
is the interval S = [0,2] (since 
is undefined for negative values of x). - The solution set for
w.r.t. 
is S = 2 π Z (see Euler's identity).