Fungrim entry: 65ccf2

Symbol: Maximum —

\mathop{\max}\limits_{x \in S} f(x)

— Maximum value of a set or function

Called with 1 argument, Maximum(S), rendered

\max\left(S\right)

, represents the maximum element of the set

S

. This operator is only defined if

S

is a subset of

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

and the maximum exists.

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

\mathop{\max}\limits_{x \in S} f(x)

, represents

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

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

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

, represents

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

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

\mathop{\max}\limits_{P(x)} f(x)

, represents

\max \left\{ f(x) : P(x) \right\}

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

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

, represents

\max \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
`Maximum`	$\mathop{\max}\limits_{x \in S} f(x)$	Maximum value of a set or function
`RR`	$\mathbb{R}$	Real numbers
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("65ccf2"),
    SymbolDefinition(Maximum, Maximum(f(x), ForElement(x, S)), "Maximum value of a set or function"),
    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(SourceForm(Maximum(f(x), ForElement(x, S))), ", rendered", Maximum(f(x), ForElement(x, S)), ", represents", Maximum(Set(f(x), ForElement(x, S))), "."),
    Description(SourceForm(Maximum(f(x), ForElement(x, S), P(x))), ", rendered", Maximum(f(x), ForElement(x, S), P(x)), ", represents", Maximum(Set(f(x), ForElement(x, S), P(x))), "."),
    Description(SourceForm(Maximum(f(x), For(x), P(x))), ", rendered", Maximum(f(x), For(x), P(x)), ", represents", Maximum(Set(f(x), For(x), P(x))), "."),
    Description(SourceForm(Maximum(f(x, y), For(Tuple(x, y)), P(x, y))), ", rendered", Maximum(f(x, y), For(Tuple(x, y)), P(x, y)), ", represents", Maximum(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