Fungrim entry: 6ec976

Symbol: Supremum —

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

— Supremum of a set or function

Supremum(S), rendered

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

, represents the supremum of the set

S

. This operator is only defined if

S

is a subset of

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

. The supremum does not need to be an element of

S

itself; in particular, for an open interval

S = \left(a, b\right)

, we have

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

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

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

, represents

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

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

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

, represents

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

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

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

, represents

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

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

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

, represents

\operatorname{sup} \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
`Supremum`	$\mathop{\operatorname{sup}}\limits_{x \in S} f(x)$	Supremum 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("6ec976"),
    SymbolDefinition(Supremum, Supremum(f(x), ForElement(x, S)), "Supremum of a set or function"),
    Description(SourceForm(Supremum(S)), ", rendered", Supremum(S), ", represents the supremum of the set", S, ".", "This operator is only defined if", S, "is a subset of", Union(RR, Set(Neg(Infinity), Pos(Infinity))), ".", "The supremum does not need to be an element of", S, "itself; in particular, for an open interval", Equal(S, OpenInterval(a, b)), ", we have", Equal(Supremum(S), b), "."),
    Description(SourceForm(Supremum(f(x), ForElement(x, S))), ", rendered", Supremum(f(x), ForElement(x, S)), ", represents", Supremum(Set(f(x), ForElement(x, S))), "."),
    Description(SourceForm(Supremum(f(x), ForElement(x, S), P(x))), ", rendered", Supremum(f(x), ForElement(x, S), P(x)), ", represents", Supremum(Set(f(x), ForElement(x, S), P(x))), "."),
    Description(SourceForm(Supremum(f(x), For(x), P(x))), ", rendered", Supremum(f(x), For(x), P(x)), ", represents", Supremum(Set(f(x), For(x), P(x))), "."),
    Description(SourceForm(Supremum(f(x, y), For(Tuple(x, y)), P(x, y))), ", rendered", Supremum(f(x, y), For(Tuple(x, y)), P(x, y)), ", represents", Supremum(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