Fungrim symbol: Solutions

Symbol: Solutions —

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

— Solution set

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

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

, represents the set of values

x \in S

satisfying

Q(x)

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

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

, represents the set of values

x \in S

satisfying

P(x)

and

Q(x)

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

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

, represents the set of values

x

satisfying

P(x)

and

Q(x)

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

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

, represents the set of tuples

\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
`Solutions`	$\mathop{\operatorname{solutions}\,}\limits_{x \in S} Q(x)$	Solution set

Source code for this entry:

Entry(ID("5862bb"),
    SymbolDefinition(Solutions, Solutions(Q(x), ForElement(x, S)), "Solution set"),
    Description(SourceForm(Solutions(Q(x), ForElement(x, S))), ", rendered", Solutions(Q(x), ForElement(x, S)), ", represents the set of values", Element(x, S), "satisfying", Q(x), "."),
    Description(SourceForm(Solutions(Q(x), ForElement(x, S), P(x))), ", rendered", Solutions(Q(x), ForElement(x, S), P(x)), ", represents the set of values", Element(x, S), "satisfying", P(x), "and", Q(x), "."),
    Description(SourceForm(Solutions(Q(x), For(x), P(x))), ", rendered", Solutions(Q(x), For(x), P(x)), ", represents the set of values", x, "satisfying", P(x), "and", Q(x), "."),
    Description(SourceForm(Solutions(Q(x, y), For(Tuple(x, y)), P(x, y))), ", rendered", Solutions(Q(x, y), For(Tuple(x, y)), P(x, y)), ", represents the set of tuples", 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 Solutions appears in 6 topics:

Chebyshev polynomials (in 3 entries)
Operators (in 1 entries)
Modular j-invariant (in 1 entries)
Lambert W-function (in 1 entries)
Inverse tangent (in 1 entries)
Imaginary unit (in 1 entries)

Entries using this symbol

The symbol Solutions appears in 8 entries: