This operator can be called with 1 or 3 arguments.
Called with 1 argument, Maximum(S), rendered max(S), represents the maximum element of the set S. This operator is only defined if S
is a subset of R∪{−∞,+∞}
and the maximum exists.
Called with 3 arguments, Maximum(f(x), x, P(x)), rendered P(x)maxf(x), represents max({f(x):P(x)}).
The argument x to this operator defines a locally bound variable. The corresponding predicate P(x)
must define the domain of x
unambiguously; that is, it must include a statement such as x∈S
where S
is a known set. More generally, x can be a collection of variables (x,y,…)
all of which become locally bound, with a corresponding predicate P(x,y,…).
Definitions:
| Fungrim symbol | Notation | Short description |
|---|---|---|
| Maximum | P(x)maxf(x) | Maximum value of a set or function |
| RR | R | Real numbers |
| Infinity | ∞ | Positive infinity |
| SetBuilder | {f(x):P(x)} | Set comprehension |
Source code for this entry:
Entry(ID("65ccf2"),
SymbolDefinition(Maximum, Maximum(f(x), x, P(x)), "Maximum value of a set or function"),
Description("This operator can be called with 1 or 3 arguments."),
Description("Called with 1 argument, ", SourceForm(Maximum(S)), ", rendered", Maximum(S), ", represents the maximum 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 maximum exists."),
Description("Called with 3 arguments, ", SourceForm(Maximum(f(x), x, P(x))), ", rendered", Maximum(f(x), x, P(x)), ", represents", Maximum(SetBuilder(f(x), x, P(x))), "."),
Description("The argument", SourceForm(x), "to this operator defines a locally bound variable.", "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.", "More generally,", SourceForm(x), "can be a collection of variables", Tuple(x, y, Ellipsis), "all of which become locally bound, with a corresponding predicate", P(x, y, Ellipsis), "."))