Fungrim entry: d0cb24

Symbol: Minimum —

\mathop{\min}\limits_{x \in S} f(x)

— Minimum value of a set or function

Minimum(S), rendered

\min\left(S\right)

, represents the minimum element of the set

S

. This operator is only defined if

S

is a subset of

\mathbb{R} \cup \left\{-\infty, +\infty\right\}

and the minimum exists.

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

\mathop{\min}\limits_{x \in S} f(x)

, represents

\min \left\{ f(x) : x \in S \right\}

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

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

, represents

\min \left\{ f(x) : x \in S \,\mathbin{\operatorname{and}}\, P(x) \right\}

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

\mathop{\min}\limits_{P(x)} f(x)

, represents

\min \left\{ f(x) : P(x) \right\}

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

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

, represents

\min \left\{ f\!\left(x, y\right) : P\!\left(x, y\right) \right\}

where

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

is a predicate defining the range of

x

and

y

, 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
`Minimum`	$\mathop{\min}\limits_{x \in S} f(x)$	Minimum value of a set or function
`RR`	$\mathbb{R}$	Real numbers
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("d0cb24"),
    SymbolDefinition(Minimum, Minimum(f(x), ForElement(x, S)), "Minimum value of a set or function"),
    Description(SourceForm(Minimum(S)), ", rendered", Minimum(S), ", represents the minimum element of the set", S, ".", "This operator is only defined if", S, "is a subset of", Union(RR, Set(Neg(Infinity), Pos(Infinity))), " and the minimum exists."),
    Description(SourceForm(Minimum(f(x), ForElement(x, S))), ", rendered", Minimum(f(x), ForElement(x, S)), ", represents", Minimum(Set(f(x), ForElement(x, S))), "."),
    Description(SourceForm(Minimum(f(x), ForElement(x, S), P(x))), ", rendered", Minimum(f(x), ForElement(x, S), P(x)), ", represents", Minimum(Set(f(x), ForElement(x, S), P(x))), "."),
    Description(SourceForm(Minimum(f(x), For(x), P(x))), ", rendered", Minimum(f(x), For(x), P(x)), ", represents", Minimum(Set(f(x), For(x), P(x))), "."),
    Description(SourceForm(Minimum(f(x, y), For(Tuple(x, y)), P(x, y))), ", rendered", Minimum(f(x, y), For(Tuple(x, y)), P(x, y)), ", represents", Minimum(Set(f(x, y), For(Tuple(x, y)), P(x, y))), "where", P(x, y), "is a predicate defining the range of", x, "and", 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 entry

Operators