Fungrim entry: 5e639e

Symbol: Sign —

\operatorname{sgn}\!\left(z\right)

— Sign function

Domain	Codomain
$z \in \mathbb{R}$	$\operatorname{sgn}\!\left(z\right) \in \left\{-1, 0, 1\right\}$
$z \in \mathbb{C} \setminus \left\{0\right\}$	$\operatorname{sgn}\!\left(z\right) \in \mathbb{T}$
$z \in \left\{\infty\right\}$	$\operatorname{sgn}\!\left(z\right) \in \left\{1\right\}$
$z \in \left\{-\infty\right\}$	$\operatorname{sgn}\!\left(z\right) \in \left\{-1\right\}$

Table data:

\left(P, Q\right)

such that

\left(P\right) \implies \left(Q\right)

Definitions:

Fungrim symbol	Notation	Short description
`Sign`	$\operatorname{sgn}\!\left(z\right)$	Sign function
`RR`	$\mathbb{R}$	Real numbers
`CC`	$\mathbb{C}$	Complex numbers
`UnitCircle`	$\mathbb{T}$	Unit circle
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("5e639e"),
    SymbolDefinition(Sign, Sign(z), "Sign function"),
    Table(TableRelation(Tuple(P, Q), Implies(P, Q)), TableHeadings(Description("Domain"), Description("Codomain")), List(Tuple(Element(z, RR), Element(Sign(z), Set(-1, 0, 1))), Tuple(Element(z, SetMinus(CC, Set(0))), Element(Sign(z), UnitCircle)), Tuple(Element(z, Set(Infinity)), Element(Sign(z), Set(1))), Tuple(Element(z, Set(Neg(Infinity))), Element(Sign(z), Set(-1))))))

Topics using this entry

Complex parts