# Solution set

> Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Solution_set
> Markdown URL: https://mediated.wiki/source/Solution_set.md
> Source: https://en.wikipedia.org/wiki/Solution_set
> Source revision: 1333817290
> License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/)

Set of values which satisfy a given set of equations

This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. Find sources: "Solution set" – news · newspapers · books · scholar · JSTOR (January 2011) (Learn how and when to remove this message)

In [mathematics](/source/Mathematics), the **solution set** of a [system of equations](/source/System_of_equations) or [inequality](/source/Inequality_(mathematics)) is the [set](/source/Set_(mathematics)) of all its solutions, that is the values that satisfy all equations and inequalities.[1] Also, the solution set or the **truth set** of a statement or a [predicate](/source/Predicate_(mathematical_logic)) is the set of all values that satisfy it.

If there is no solution, the solution set is the [empty set](/source/Empty_set).[2]

## Examples

- The solution set of the single equation x = 0 {\displaystyle x=0} is the [singleton set](/source/Singleton_set) { 0 } {\displaystyle \{0\}} .

- Since there do not exist numbers x {\displaystyle x} and y {\displaystyle y} making the two equations { x + 2 y = 3 , x + 2 y = − 3 {\displaystyle {\begin{cases}x+2y=3,&\\x+2y=-3\end{cases}}} simultaneously true, the solution set of this system is the [empty set](/source/Empty_set) ∅ {\displaystyle \emptyset } .

- The solution set of a [constrained optimization problem](/source/Constrained_optimization_problem) is its [feasible region](/source/Feasible_region).

- The truth set of the predicate P ( n ) : n i s e v e n {\displaystyle P(n):n\mathrm {\ is\ even} } is { 2 , 4 , 6 , 8 , … } {\displaystyle \{2,4,6,8,\ldots \}} .

## Remarks

In [algebraic geometry](/source/Algebraic_geometry), solution sets are called [algebraic sets](/source/Algebraic_set) if there are no inequalities. Over the [reals](/source/Real_number), and with inequalities, there are called [semialgebraic sets](/source/Semialgebraic_set).

## Other meanings

More generally, the **solution set** to an arbitrary collection *E* of [relations](/source/Relation_(mathematics)) (*Ei*) (*i* varying in some index set *I*) for a collection of unknowns ( x j ) j ∈ J {\displaystyle {(x_{j})}_{j\in J}} , supposed to take values in respective spaces ( X j ) j ∈ J {\displaystyle {(X_{j})}_{j\in J}} , is the set *S* of all solutions to the relations *E*, where a solution x ( k ) {\displaystyle x^{(k)}} is a family of values ( x j ( k ) ) j ∈ J ∈ ∏ j ∈ J X j {\textstyle {\left(x_{j}^{(k)}\right)}_{j\in J}\in \prod _{j\in J}X_{j}} such that substituting ( x j ) j ∈ J {\displaystyle {\left(x_{j}\right)}_{j\in J}} by 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](/source/Predicate_(mathematics)), the collection *E* is their [logical conjunction](/source/Logical_conjunction), and the solution set is the [inverse image](/source/Inverse_image) of the Boolean value *true* by the associated [Boolean-valued function](/source/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.

### Examples

- The solution set for *E* = { *x*+*y* = 0 } with respect to ( x , y ) ∈ R 2 {\displaystyle (x,y)\in \mathbb {R} ^{2}} is *S* = { (*a*,−*a*) : *a* ∈ **R** }.

- The solution set for *E* = { *x*+*y* = 0 } with respect to x ∈ 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](/source/Parameter) on which the equation, and therefore the solution set, depends.)

- The solution set for E = { x ≤ 4 } {\displaystyle E=\{{\sqrt {x}}\leq 4\}} with respect to x ∈ R {\displaystyle x\in \mathbb {R} } is the interval *S* = [0,16] (since x {\displaystyle {\sqrt {x}}} is undefined for negative values of *x*).

- The solution set for E = { e i x = 1 } {\displaystyle E=\{e^{ix}=1\}} with respect to x ∈ C {\displaystyle x\in \mathbb {C} } is *S* = 2π**Z** (see [Euler's identity](/source/Euler's_identity)).

## See also

- [Equation solving](/source/Equation_solving)

- [Extraneous and missing solutions](/source/Extraneous_and_missing_solutions)

- [Equaliser (mathematics)](/source/Equaliser_(mathematics))

## References

1. **[^](#cite_ref-1)** ["Definition of SOLUTION SET"](https://www.merriam-webster.com/dictionary/solution+set). *www.merriam-webster.com*. Retrieved 2024-08-14.

1. **[^](#cite_ref-2)** ["Systems of Linear Equations"](https://textbooks.math.gatech.edu/ila/systems-of-eqns.html). *textbooks.math.gatech.edu*. Retrieved 2024-08-14.

---
Adapted from the Wikipedia article [Solution set](https://en.wikipedia.org/wiki/Solution_set) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Solution_set?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.