Fungrim home page

Fungrim entry: 6ec976

Symbol: Supremum supxSf(x)\mathop{\operatorname{sup}}\limits_{x \in S} f(x) Supremum of a set or function
Supremum(S), rendered sup(S)\operatorname{sup}\left(S\right), represents the supremum of the set SS. This operator is only defined if SS is a subset of R{,+}\mathbb{R} \cup \left\{-\infty, +\infty\right\}. The supremum does not need to be an element of SS itself; in particular, for an open interval S=(a,b)S = \left(a, b\right), we have sup(S)=b\operatorname{sup}\left(S\right) = b.
Supremum(f(x), ForElement(x, S)), rendered supxSf(x)\mathop{\operatorname{sup}}\limits_{x \in S} f(x), represents sup{f(x):xS}\operatorname{sup} \left\{ f(x) : x \in S \right\}.
Supremum(f(x), ForElement(x, S), P(x)), rendered supxS,P(x)f(x)\mathop{\operatorname{sup}}\limits_{x \in S,\,P(x)} f(x), represents sup{f(x):xSandP(x)}\operatorname{sup} \left\{ f(x) : x \in S \,\mathbin{\operatorname{and}}\, P(x) \right\}.
Supremum(f(x), For(x), P(x)), rendered supP(x)f(x)\mathop{\operatorname{sup}}\limits_{P(x)} f(x), represents sup{f(x):P(x)}\operatorname{sup} \left\{ f(x) : P(x) \right\}.
Supremum(f(x, y), For(Tuple(x, y)), P(x, y)), rendered supP(x,y)f ⁣(x,y)\mathop{\operatorname{sup}}\limits_{P\left(x, y\right)} f\!\left(x, y\right), represents sup{f ⁣(x,y):P ⁣(x,y)}\operatorname{sup} \left\{ f\!\left(x, y\right) : P\!\left(x, y\right) \right\} where P ⁣(x,y)P\!\left(x, y\right) is a predicate defining the range of xx and yy, and similarly for any number n2n \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)P(x) must define the domain of xx unambiguously; that is, it must include a statement such as xSx \in S where SS 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 ⁣(x,y)P\!\left(x, y\right), P ⁣(x,y,z)P\!\left(x, y, z\right), etc.
Definitions:
Fungrim symbol Notation Short description
SupremumsupxSf(x)\mathop{\operatorname{sup}}\limits_{x \in S} f(x) Supremum of a set or function
RRR\mathbb{R} Real numbers
Infinity\infty Positive infinity
OpenInterval(a,b)\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

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2020-01-31 18:09:28.494564 UTC