HOME

TheInfoList



OR:

In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the solution set of a system of equations or inequality is the
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
of all its solutions, that is the values that satisfy all equations and inequalities. Also, the solution set or the truth set of a statement or a predicate is the set of all values that satisfy it. If there is no solution, the solution set is the
empty set In mathematics, the empty set or void set is the unique Set (mathematics), set having no Element (mathematics), elements; its size or cardinality (count of elements in a set) is 0, zero. Some axiomatic set theories ensure that the empty set exi ...
.


Examples

* The solution set of the single equation x=0 is the
singleton set In mathematics, a singleton (also known as a unit set or one-point set) is a set with exactly one element. For example, the set \ is a singleton whose single element is 0. Properties Within the framework of Zermelo–Fraenkel set theory, the a ...
\. * Since there do not exist numbers x and y making the two equations \begin x + 2y = 3,&\\ x + 2y = -3 \end simultaneously true, the solution set of this system is the * The solution set of a constrained optimization problem is its
feasible region In mathematical optimization and computer science, a feasible region, feasible set, 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, ...
. * The truth set of the predicate P(n): n \mathrm is \.


Remarks

In
algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...
, solution sets are called
algebraic set Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. ...
s 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
relation Relation or relations may refer to: General uses * International relations, the study of interconnection of politics, economics, and law on a global level * Interpersonal relationship, association or acquaintance between two or more people * ...
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 x^ is a family of values _\in \prod_ X_j such that substituting _ by x^ 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 In logic, mathematics and linguistics, ''and'' (\wedge) is the Truth function, truth-functional operator of conjunction or logical conjunction. The logical connective of this operator is typically represented as \wedge or \& or K (prefix) or ...
, and the solution set is the
inverse image In mathematics, for a function f: X \to Y, the image of an input value x is the single output value produced by f when passed x. The preimage of an output value y is the set of input values that produce y. More generally, evaluating f at each ...
of the boolean value ''true'' by the associated
boolean-valued function A Boolean-valued function (sometimes called 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, i.e. a generic two-element set, (for example B = ), whose elements ar ...
.) 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'' = 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 In mathematics, Euler's identity (also known as Euler's equation) is the Equality (mathematics), equality e^ + 1 = 0 where :e is E (mathematical constant), Euler's number, the base of natural logarithms, :i is the imaginary unit, which by definit ...
).


See also

*
Equation solving In mathematics, to solve an equation is to find its solutions, which are the values (numbers, functions, sets, etc.) that fulfill the condition stated by the equation, consisting generally of two expressions related by an equals sign. When s ...
* Extraneous and missing solutions *
Equaliser (mathematics) In mathematics, an equaliser is a set of arguments where two or more functions have equal values. An equaliser is the solution set of an equation. In certain contexts, a difference kernel is the equaliser of exactly two functions. Definition ...


References

{{DEFAULTSORT:Solution Set Equations