Fungrim symbol: UniqueSolution

Symbol: UniqueSolution —

\mathop{\operatorname{solution*}\,}\limits_{x \in S} Q(x)

— Unique solution

UniqueSolution(Q(x), ForElement(x, S)), rendered

\mathop{\operatorname{solution*}\,}\limits_{x \in S} Q(x)

, represents the unique value

x \in S

satisfying

Q(x)

This operation is undefined if such a value does not exist or is not unique.

UniqueSolution(Q(x), ForElement(x, S), P(x)), rendered

\mathop{\operatorname{solution*}\,}\limits_{x \in S,\,P(x)} Q(x)

, represents the unique value

x \in S

satisfying

P(x)

and

Q(x)

UniqueSolution(Q(x), For(x), P(x)), rendered

\mathop{\operatorname{solution*}\,}\limits_{P(x)} Q(x)

, represents the unique value

x

satisfying

P(x)

and

Q(x)

UniqueSolution(Q(x, y), For(Tuple(x, y)), P(x, y)), rendered

\mathop{\operatorname{solution*}\,}\limits_{P\left(x, y\right)} Q\!\left(x, y\right)

, represents the unique tuple

\left(x, y\right)

satisfying

P\!\left(x, y\right)

and

Q\!\left(x, y\right)

, and similarly for any number

n \ge 2

of variables.

The special expression For(x) or ForElement(x, S) declares x as a locally bound variable within the scope of the arguments to this operator. If For(x) is used instead of ForElement(x, S), the corresponding predicate

P(x)

must define the domain of

x

unambiguously; that is, it must include a statement such as

x \in S

where

S

is a known set. Similarly, For(Tuple(x, y)), For(Tuple(x, y, z)), etc. defines multiple locally bound variables which must be accompanied by a multivariate predicate

P\!\left(x, y\right)

P\!\left(x, y, z\right)

, etc.

Definitions:

Fungrim symbol	Notation	Short description
`UniqueSolution`	$\mathop{\operatorname{solution*}\,}\limits_{x \in S} Q(x)$	Unique solution

Source code for this entry:

Entry(ID("f5ae93"),
    SymbolDefinition(UniqueSolution, UniqueSolution(Q(x), ForElement(x, S)), "Unique solution"),
    Description(SourceForm(UniqueSolution(Q(x), ForElement(x, S))), ", rendered", UniqueSolution(Q(x), ForElement(x, S)), ", represents the unique value", Element(x, S), "satisfying", Q(x), "."),
    Description("This operation is undefined if such a value does not exist or is not unique."),
    Description(SourceForm(UniqueSolution(Q(x), ForElement(x, S), P(x))), ", rendered", UniqueSolution(Q(x), ForElement(x, S), P(x)), ", represents the unique value", Element(x, S), "satisfying", P(x), "and", Q(x), "."),
    Description(SourceForm(UniqueSolution(Q(x), For(x), P(x))), ", rendered", UniqueSolution(Q(x), For(x), P(x)), ", represents the unique value", x, "satisfying", P(x), "and", Q(x), "."),
    Description(SourceForm(UniqueSolution(Q(x, y), For(Tuple(x, y)), P(x, y))), ", rendered", UniqueSolution(Q(x, y), For(Tuple(x, y)), P(x, y)), ", represents the unique tuple", Tuple(x, y), "satisfying", P(x, y), "and", Q(x, y), ", and similarly for any number", GreaterEqual(n, 2), "of variables."),
    Description("The special expression", SourceForm(For(x)), "or", SourceForm(ForElement(x, S)), "declares", SourceForm(x), "as a locally bound variable within the scope of the arguments to this operator. ", "If", SourceForm(For(x)), "is used instead of", SourceForm(ForElement(x, S)), ", the corresponding predicate", P(x), "must define the domain of", x, "unambiguously; that is, it must include a statement such as", Element(x, S), "where", S, "is a known set. Similarly,", SourceForm(For(Tuple(x, y))), ", ", SourceForm(For(Tuple(x, y, z))), ", etc.", "defines multiple locally bound variables which must be accompanied by a multivariate predicate", P(x, y), ", ", P(x, y, z), ", etc."))

Topics using this symbol

The symbol UniqueSolution appears in 5 topics:

Lambert W-function (in 2 entries)
Riemann hypothesis (in 1 entries)
Operators (in 1 entries)
Landau's function (in 1 entries)
Inverse tangent (in 1 entries)

Entries using this symbol

The symbol UniqueSolution appears in 5 entries: