Fungrim symbol: OpenInterval

Symbol: OpenInterval —

\left(a, b\right)

— Open interval

►OpenInterval(a, b) —

\left(a, b\right)

— Represents

\left\{ x : x \in \mathbb{R} \cup \left\{-\infty, \infty\right\} \,\mathbin{\operatorname{and}}\, a < x < b \right\}

►OpenInterval(0, 1) —

\left(0, 1\right)

— Represents the open unit interval.

►OpenInterval(1, 1) —

\left(1, 1\right)

— Represents the empty set.

►OpenInterval(Neg(Infinity), 0) —

\left(-\infty, 0\right)

— Represents half the extended real line (excluding minus infinity and zero).

►OpenInterval(1, -1) —

\left(1, -1\right)

— Represents the empty set. Note: potentially confusing rendering.

►Add(1, Mul(OpenInterval(0, 1), ConstI)) —

1 + \left(0, 1\right) i

— Represents a set of points in the complex plane. OpenInterval(a, b) should only be used with extended real number

a

and

b

as endpoints, but line segments in the complex plane can be constructed by applying arithmetic operations to a set of real numbers (acting pointwise).

►Add(OpenInterval(1, 4), Mul(OpenInterval(0, 1), ConstI)) —

\left(1, 4\right) + \left(0, 1\right) i

— Represents a rectangle in the complex plane.

Definitions:

Fungrim symbol	Notation	Short description
`OpenInterval`	$\left(a, b\right)$	Open interval
`RR`	$\mathbb{R}$	Real numbers
`Infinity`	$\infty$	Positive infinity
`ConstI`	$i$	Imaginary unit

Source code for this entry:

Entry(ID("3fe68f"),
    SymbolDefinition(OpenInterval, OpenInterval(a, b), "Open interval"),
    CodeExample(OpenInterval(a, b), "Represents", Set(x, ForElement(x, Union(RR, Set(Neg(Infinity), Infinity))), Less(a, x, b)), "."),
    CodeExample(OpenInterval(0, 1), "Represents the open unit interval."),
    CodeExample(OpenInterval(1, 1), "Represents the empty set."),
    CodeExample(OpenInterval(Neg(Infinity), 0), "Represents half the extended real line (excluding minus infinity and zero)."),
    CodeExample(OpenInterval(1, -1), "Represents the empty set.", " Note: potentially confusing rendering."),
    CodeExample(Add(1, Mul(OpenInterval(0, 1), ConstI)), "Represents a set of points in the complex plane. ", SourceForm(OpenInterval(a, b)), "should only be used with extended real number", a, "and", b, "as endpoints, but line segments in the complex plane can be constructed by applying arithmetic operations to a set of real numbers (acting pointwise)."),
    CodeExample(Add(OpenInterval(1, 4), Mul(OpenInterval(0, 1), ConstI)), "Represents a rectangle in the complex plane. "))

Topics using this symbol

The symbol OpenInterval appears in 35 topics:

Carlson symmetric elliptic integrals (in 43 entries)
Specific values of Carlson symmetric elliptic integrals (in 17 entries)
Digamma function (in 14 entries)
Series representations of Carlson symmetric elliptic integrals (in 13 entries)
Square roots (in 8 entries)
Lambert W-function (in 8 entries)
Barnes G-function (in 8 entries)
Bounds and inequalities for the gamma function (in 7 entries)
Legendre elliptic integrals (in 6 entries)
Bessel functions (in 6 entries)
Natural logarithm (in 5 entries)
Halphen's constant (in 5 entries)
Specific values of Jacobi theta functions (in 4 entries)
Operators (in 4 entries)
Inverse tangent (in 4 entries)
Integrals of Jacobi theta functions (in 4 entries)
Modular lambda function (in 3 entries)
Gamma function (in 3 entries)
Sinc function (in 2 entries)
Legendre polynomials (in 2 entries)
Eisenstein series (in 2 entries)
Arithmetic-geometric mean (in 2 entries)
Zeros of the Riemann zeta function (in 1 entries)
Totient function (in 1 entries)
Riemann zeta function (in 1 entries)
Riemann hypothesis (in 1 entries)
Numbers and infinities (in 1 entries)
Landau's function (in 1 entries)
Hurwitz zeta function (in 1 entries)
Golden ratio (in 1 entries)
Factorials and binomial coefficients (in 1 entries)
Exponential function (in 1 entries)
Complex parts (in 1 entries)
Chebyshev polynomials (in 1 entries)
Bernoulli numbers and polynomials (in 1 entries)

Entries using this symbol

The symbol OpenInterval appears in 169 entries: