Fungrim entry: f7ce46

Symbol: Zeros —

\mathop{\operatorname{zeros}\,}\limits_{x \in S} f(x)

— Zeros (roots) of function

Zeros(f(x), ForElement(x, S)), rendered

\mathop{\operatorname{zeros}\,}\limits_{x \in S} f(x)

, represents the set of values

x \in S

satisfying

f(x) = 0

Zeros(f(x), ForElement(x, S), P(x)), rendered

\mathop{\operatorname{zeros}\,}\limits_{x \in S,\,P(x)} f(x)

, represents the set of values

x \in S

satisfying

P(x)

and

f(x) = 0

Zeros(f(x), For(x), P(x)), rendered

\mathop{\operatorname{zeros}\,}\limits_{P(x)} f(x)

, represents the set of values

x

satisfying

P(x)

and

f(x) = 0

Zeros(f(x, y), For(Tuple(x, y)), P(x, y)), rendered

\mathop{\operatorname{zeros}\,}\limits_{P\left(x, y\right)} f\!\left(x, y\right)

, represents the set of tuples

\left(x, y\right)

satisfying

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

and

f\!\left(x, y\right) = 0

, 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
`Zeros`	$\mathop{\operatorname{zeros}\,}\limits_{x \in S} f(x)$	Zeros (roots) of function

Source code for this entry:

Entry(ID("f7ce46"),
    SymbolDefinition(Zeros, Zeros(f(x), ForElement(x, S)), "Zeros (roots) of function"),
    Description(SourceForm(Zeros(f(x), ForElement(x, S))), ", rendered", Zeros(f(x), ForElement(x, S)), ", represents the set of values", Element(x, S), "satisfying", Equal(f(x), 0), "."),
    Description(SourceForm(Zeros(f(x), ForElement(x, S), P(x))), ", rendered", Zeros(f(x), ForElement(x, S), P(x)), ", represents the set of values", Element(x, S), "satisfying", P(x), "and", Equal(f(x), 0), "."),
    Description(SourceForm(Zeros(f(x), For(x), P(x))), ", rendered", Zeros(f(x), For(x), P(x)), ", represents the set of values", x, "satisfying", P(x), "and", Equal(f(x), 0), "."),
    Description(SourceForm(Zeros(f(x, y), For(Tuple(x, y)), P(x, y))), ", rendered", Zeros(f(x, y), For(Tuple(x, y)), P(x, y)), ", represents the set of tuples", Tuple(x, y), "satisfying", P(x, y), "and", Equal(f(x, y), 0), ", 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 entry

Operators