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# Fungrim entry: 6ec976

Symbol: Supremum $\mathop{\operatorname{sup}}\limits_{P\left(x\right)} f\!\left(x\right)$ Supremum of a set or function
This operator can be called with 1 or 3 arguments.
Called with 1 argument, 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 $\operatorname{sup}\left(S\right) = b$.
Called with 3 arguments, Supremum(f(x), Var(x), P(x)), rendered $\mathop{\operatorname{sup}}\limits_{P\left(x\right)} f\!\left(x\right)$, represents $\operatorname{sup} \left\{ f\!\left(x\right) : P\!\left(x\right) \right\}$ where $P\!\left(x\right)$ is a predicate defining the range of $x$.
Supremum(f(x, y), Var(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 Var(x) declares x as a locally bound variable within the scope of the arguments to this operator. The corresponding predicate $P\!\left(x\right)$ 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, Var(x, y), Var(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$\mathop{\operatorname{sup}}\limits_{P\left(x\right)} f\!\left(x\right)$ Supremum of a set or function
RR$\mathbb{R}$ Real numbers
Infinity$\infty$ Positive infinity
OpenInterval$\left(a, b\right)$ Open interval
SetBuilder$\left\{ f\!\left(x\right) : P\!\left(x\right) \right\}$ Set comprehension
Source code for this entry:
Entry(ID("6ec976"),
SymbolDefinition(Supremum, Supremum(f(x), Var(x), P(x)), "Supremum of a set or function"),
Description("This operator can be called with 1 or 3 arguments."),
Description("Called with 1 argument, ", 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("Called with 3 arguments, ", SourceForm(Supremum(f(x), Var(x), P(x))), ", rendered", Supremum(f(x), Var(x), P(x)), ", represents", Supremum(SetBuilder(f(x), x, P(x))), "where", P(x), "is a predicate defining the range of", x, "."),
Description(SourceForm(Supremum(f(x, y), Var(x, y), P(x, y))), ", rendered", Supremum(f(x, y), Var(x, y), P(x, y)), ", represents", Supremum(SetBuilder(f(x, y), x, 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(Var(x)), "declares", SourceForm(x), "as a locally bound variable within the scope of the arguments to this operator. ", "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(Var(x, y)), ", ", SourceForm(Var(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.

2019-09-19 20:12:49.583742 UTC