Fungrim entry: bbeb35

Symbol: Infimum —

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

— Infimum of a set or function

Infimum(S), rendered

\operatorname{inf}\left(S\right)

, represents the infimum of the set

S

. This operator is only defined if

S

is a subset of

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

. The infimum does not need to be an element of

S

itself; in particular, for an open interval

S = \left(a, b\right)

, we have

\operatorname{inf}\left(S\right) = b

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

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

, represents

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

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

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

, represents

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

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

\mathop{\operatorname{inf}}\limits_{P(x)} f(x)

, represents

\operatorname{inf} \left\{ f(x) : P(x) \right\}

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

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

, represents

\operatorname{inf} \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
`Infimum`	$\mathop{\operatorname{inf}}\limits_{x \in S} f(x)$	Infimum of a set or function
`RR`	$\mathbb{R}$	Real numbers
`Infinity`	$\infty$	Positive infinity
`OpenInterval`	$\left(a, b\right)$	Open interval

Source code for this entry:

Entry(ID("bbeb35"),
    SymbolDefinition(Infimum, Infimum(f(x), ForElement(x, S)), "Infimum of a set or function"),
    Description(SourceForm(Infimum(S)), ", rendered", Infimum(S), ", represents the infimum of the set", S, ".", "This operator is only defined if", S, "is a subset of", Union(RR, Set(Neg(Infinity), Pos(Infinity))), ".", "The infimum does not need to be an element of", S, "itself; in particular, for an open interval", Equal(S, OpenInterval(a, b)), ", we have", Equal(Infimum(S), b), "."),
    Description(SourceForm(Infimum(f(x), ForElement(x, S))), ", rendered", Infimum(f(x), ForElement(x, S)), ", represents", Infimum(Set(f(x), ForElement(x, S))), "."),
    Description(SourceForm(Infimum(f(x), ForElement(x, S), P(x))), ", rendered", Infimum(f(x), ForElement(x, S), P(x)), ", represents", Infimum(Set(f(x), ForElement(x, S), P(x))), "."),
    Description(SourceForm(Infimum(f(x), For(x), P(x))), ", rendered", Infimum(f(x), For(x), P(x)), ", represents", Infimum(Set(f(x), For(x), P(x))), "."),
    Description(SourceForm(Infimum(f(x, y), For(Tuple(x, y)), P(x, y))), ", rendered", Infimum(f(x, y), For(Tuple(x, y)), P(x, y)), ", represents", Infimum(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