Numbers and infinities

Table of contents: Numbers - Infinities - Ranges and intervals

Numbers

Standard number domains

Symbol: ZZ —

\mathbb{Z}

— Integers

Represents the set of integers.

Definitions:

Fungrim symbol	Notation	Short description
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("298e9e"),
    SymbolDefinition(ZZ, ZZ, "Integers"),
    Description("Represents the set of integers."))

7be5dc

Symbol: QQ —

\mathbb{Q}

— Rational numbers

Represents the set of rational numbers.

Definitions:

Fungrim symbol	Notation	Short description
`QQ`	$\mathbb{Q}$	Rational numbers

Source code for this entry:

Entry(ID("7be5dc"),
    SymbolDefinition(QQ, QQ, "Rational numbers"),
    Description("Represents the set of rational numbers."))

bfe358

Symbol: RR —

\mathbb{R}

— Real numbers

Represents the set of real numbers.

Definitions:

Fungrim symbol	Notation	Short description
`RR`	$\mathbb{R}$	Real numbers

Source code for this entry:

Entry(ID("bfe358"),
    SymbolDefinition(RR, RR, "Real numbers"),
    Description("Represents the set of real numbers."))

0deea6

Symbol: CC —

\mathbb{C}

— Complex numbers

Represents the set of complex numbers.

Definitions:

Fungrim symbol	Notation	Short description
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("0deea6"),
    SymbolDefinition(CC, CC, "Complex numbers"),
    Description("Represents the set of complex numbers."))

4fd123

\mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C}

TeX:

\mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`ZZ`	$\mathbb{Z}$	Integers
`QQ`	$\mathbb{Q}$	Rational numbers
`RR`	$\mathbb{R}$	Real numbers
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("4fd123"),
    Formula(Subset(ZZ, QQ, RR, CC)))

Rational numbers

c01d22

\mathbb{Q} = \left\{ \frac{p}{q} : p \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; q \in \mathbb{Z} \setminus \left\{0\right\} \right\}

TeX:

\mathbb{Q} = \left\{ \frac{p}{q} : p \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; q \in \mathbb{Z} \setminus \left\{0\right\} \right\}

Definitions:

Fungrim symbol	Notation	Short description
`QQ`	$\mathbb{Q}$	Rational numbers
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("c01d22"),
    Formula(Equal(QQ, Set(Div(p, q), For(Tuple(p, q)), And(Element(p, ZZ), Element(q, SetMinus(ZZ, Set(0))))))))

81c491

\sqrt{2} \notin \mathbb{Q}

TeX:

\sqrt{2} \notin \mathbb{Q}

Definitions:

Fungrim symbol	Notation	Short description
`Sqrt`	$\sqrt{z}$	Principal square root
`QQ`	$\mathbb{Q}$	Rational numbers

Source code for this entry:

Entry(ID("81c491"),
    Formula(NotElement(Sqrt(2), QQ)))

0c838a

\pi \notin \mathbb{Q}

TeX:

\pi \notin \mathbb{Q}

Definitions:

Fungrim symbol	Notation	Short description
`Pi`	$\pi$	The constant pi (3.14...)
`QQ`	$\mathbb{Q}$	Rational numbers

Source code for this entry:

Entry(ID("0c838a"),
    Formula(NotElement(Pi, QQ)))

Complex numbers

Related topics: Imaginary unit, Complex plane

88ad6f

i \in \mathbb{C}

TeX:

i \in \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`ConstI`	$i$	Imaginary unit
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("88ad6f"),
    Formula(Element(ConstI, CC)))

72cef9

i = \sqrt{-1}

TeX:

i = \sqrt{-1}

Definitions:

Fungrim symbol	Notation	Short description
`ConstI`	$i$	Imaginary unit
`Sqrt`	$\sqrt{z}$	Principal square root

Source code for this entry:

Entry(ID("72cef9"),
    Formula(Equal(ConstI, Sqrt(-1))))

a08fb9

i \notin \mathbb{R}

TeX:

i \notin \mathbb{R}

Definitions:

Fungrim symbol	Notation	Short description
`ConstI`	$i$	Imaginary unit
`RR`	$\mathbb{R}$	Real numbers

Source code for this entry:

Entry(ID("a08fb9"),
    Formula(NotElement(ConstI, RR)))

77ef0c

\mathbb{C} = \left\{ x + y i : x \in \mathbb{R} \;\mathbin{\operatorname{and}}\; y \in \mathbb{R} \right\}

TeX:

\mathbb{C} = \left\{ x + y i : x \in \mathbb{R} \;\mathbin{\operatorname{and}}\; y \in \mathbb{R} \right\}

Definitions:

Fungrim symbol	Notation	Short description
`CC`	$\mathbb{C}$	Complex numbers
`ConstI`	$i$	Imaginary unit
`RR`	$\mathbb{R}$	Real numbers

Source code for this entry:

Entry(ID("77ef0c"),
    Formula(Equal(CC, Set(Add(x, Mul(y, ConstI)), For(Tuple(x, y)), And(Element(x, RR), Element(y, RR))))))

Algebraic numbers

be9c83

Symbol: AlgebraicNumbers —

\overline{\mathbb{Q}}

— Algebraic numbers

Represents the set of algebraic numbers.

Definitions:

Fungrim symbol	Notation	Short description
`AlgebraicNumbers`	$\overline{\mathbb{Q}}$	Algebraic numbers

Source code for this entry:

Entry(ID("be9c83"),
    SymbolDefinition(AlgebraicNumbers, AlgebraicNumbers, "Algebraic numbers"),
    Description("Represents the set of algebraic numbers."))

aa6b07

\overline{\mathbb{Q}} = \left\{ z : z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \left(f(z) = 0 \;\text{ for some } f \in \mathbb{Z}[\text{x}_{1}] \setminus \left\{0\right\}\right) \right\}

TeX:

\overline{\mathbb{Q}} = \left\{ z : z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \left(f(z) = 0 \;\text{ for some } f \in \mathbb{Z}[\text{x}_{1}] \setminus \left\{0\right\}\right) \right\}

Definitions:

Fungrim symbol	Notation	Short description
`AlgebraicNumbers`	$\overline{\mathbb{Q}}$	Algebraic numbers
`CC`	$\mathbb{C}$	Complex numbers
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("aa6b07"),
    Formula(Equal(AlgebraicNumbers, Set(z, ForElement(z, CC), Parentheses(Exists(Equal(CallIndeterminate(f, XX(1), z), 0), ForElement(f, SetMinus(Polynomials(ZZ, XX(1)), Set(0)))))))))

e5a04c

\mathbb{Z} \subset \mathbb{Q} \subset \overline{\mathbb{Q}} \subset \mathbb{C}

TeX:

\mathbb{Z} \subset \mathbb{Q} \subset \overline{\mathbb{Q}} \subset \mathbb{C}

Definitions:

Fungrim symbol	Notation	Short description
`ZZ`	$\mathbb{Z}$	Integers
`QQ`	$\mathbb{Q}$	Rational numbers
`AlgebraicNumbers`	$\overline{\mathbb{Q}}$	Algebraic numbers
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("e5a04c"),
    Formula(Subset(ZZ, QQ, AlgebraicNumbers, CC)))

24c179

\sqrt{2} \in \overline{\mathbb{Q}}

TeX:

\sqrt{2} \in \overline{\mathbb{Q}}

Definitions:

Fungrim symbol	Notation	Short description
`Sqrt`	$\sqrt{z}$	Principal square root
`AlgebraicNumbers`	$\overline{\mathbb{Q}}$	Algebraic numbers

Source code for this entry:

Entry(ID("24c179"),
    Formula(Element(Sqrt(2), AlgebraicNumbers)))

cd8a07

i \in \overline{\mathbb{Q}}

TeX:

i \in \overline{\mathbb{Q}}

Definitions:

Fungrim symbol	Notation	Short description
`ConstI`	$i$	Imaginary unit
`AlgebraicNumbers`	$\overline{\mathbb{Q}}$	Algebraic numbers

Source code for this entry:

Entry(ID("cd8a07"),
    Formula(Element(ConstI, AlgebraicNumbers)))

155575

\pi \notin \overline{\mathbb{Q}}

TeX:

\pi \notin \overline{\mathbb{Q}}

Definitions:

Fungrim symbol	Notation	Short description
`Pi`	$\pi$	The constant pi (3.14...)
`AlgebraicNumbers`	$\overline{\mathbb{Q}}$	Algebraic numbers

Source code for this entry:

Entry(ID("155575"),
    Formula(NotElement(Pi, AlgebraicNumbers)))

Infinities

b738b1

Symbol: Infinity —

\infty

— Positive infinity

This formal symbol represents a quantity larger than any real number. We define

+\infty = \infty

Multiplication of

\infty

by a nonzero complex number represents an infinite limit with the given direction in the complex plane. In particular,

-\infty

i \infty

and

-i \infty

are frequently used.

The set

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

is known as the extended real line.

Definitions:

Fungrim symbol	Notation	Short description
`Infinity`	$\infty$	Positive infinity
`ConstI`	$i$	Imaginary unit
`RR`	$\mathbb{R}$	Real numbers

Source code for this entry:

Entry(ID("b738b1"),
    SymbolDefinition(Infinity, Infinity, "Positive infinity"),
    Description("This formal symbol represents a quantity larger than any real number. We define", Equal(Pos(Infinity), Infinity), "."),
    Description("Multiplication of", Infinity, "by a nonzero complex number represents an infinite limit with the given direction in the complex plane.", "In particular,", Neg(Infinity), ",", Mul(ConstI, Infinity), "and", Mul(Neg(ConstI), Infinity), "are frequently used."),
    Description("The set", Union(RR, Set(Infinity, Neg(Infinity))), "is known as the extended real line."))

486ab2

Symbol: UnsignedInfinity —

{\tilde \infty}

— Unsigned infinity

This formal symbol represents a quantity with infinite magnitude and undefined sign.

It is typically used to represent the value of meromorphic functions at poles.

The set

\mathbb{C} \cup \left\{{\tilde \infty}\right\}

represents the complex Riemann sphere.

Definitions:

Fungrim symbol	Notation	Short description
`UnsignedInfinity`	${\tilde \infty}$	Unsigned infinity
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("486ab2"),
    SymbolDefinition(UnsignedInfinity, UnsignedInfinity, "Unsigned infinity"),
    Description("This formal symbol represents a quantity with infinite magnitude and undefined sign."),
    Description("It is typically used to represent the value of meromorphic functions at poles."),
    Description("The set", Union(CC, Set(UnsignedInfinity)), "represents the complex Riemann sphere."))

Ranges and intervals

03fbae

Symbol: ZZGreaterEqual —

\mathbb{Z}_{\ge n}

— Integers greater than or equal to n

Definitions:

Fungrim symbol	Notation	Short description
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("03fbae"),
    SymbolDefinition(ZZGreaterEqual, ZZGreaterEqual(n), "Integers greater than or equal to n"))

2a52af

Symbol: ZZLessEqual —

\mathbb{Z}_{\le n}

— Integers less than or equal to n

This symbol may be rendered differently when

n

is a concrete value, for example:

\{-3, -4, \ldots\}

Definitions:

Fungrim symbol	Notation	Short description
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n

Source code for this entry:

Entry(ID("2a52af"),
    SymbolDefinition(ZZLessEqual, ZZLessEqual(n), "Integers less than or equal to n"),
    Description("This symbol may be rendered differently when", n, "is a concrete value, for example: ", ZZLessEqual(-3)))

00b82b

Symbol: Range —

\{a, a + 1, \ldots, b\}

— Integers between given endpoints

►Range(a, b) —

\{a, a + 1, \ldots, b\}

— Given

a \in \mathbb{Z}

and

b \in \mathbb{Z}

, represents

\left\{ n : n \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, a \le n \le b \right\}

►Range(a, b) —

\{a, a + 1, \ldots, b\}

— Given

a \in \mathbb{Z}

and

b \in \mathbb{Z}

, is equivalent to Set(n, For(n, a, b)).

►Range(3, 3) —

\{3, 4, \ldots, 3\}

— Represents the singleton set {3}. Note: potentially confusing rendering.

►Range(3, 2) —

\{3, 4, \ldots, 2\}

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

Definitions:

Fungrim symbol	Notation	Short description
`Range`	$\{a, a + 1, \ldots, b\}$	Integers between given endpoints
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("00b82b"),
    SymbolDefinition(Range, Range(a, b), "Integers between given endpoints"),
    CodeExample(Range(a, b), "Given", Element(a, ZZ), "and", Element(b, ZZ), ", represents", Set(n, ForElement(n, ZZ), LessEqual(a, n, b)), "."),
    CodeExample(Range(a, b), "Given", Element(a, ZZ), "and", Element(b, ZZ), ", is equivalent to", SourceForm(Set(n, For(n, a, b))), "."),
    CodeExample(Range(3, 3), "Represents the singleton set", Set(3), ".", " Note: potentially confusing rendering."),
    CodeExample(Range(3, 2), "Represents the empty set.", " Note: potentially confusing rendering."))

12d5ab

Symbol: ClosedInterval —

\left[a, b\right]

— Closed interval

►ClosedInterval(a, b) —

\left[a, b\right]

— Represents

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

►ClosedInterval(0, 1) —

\left[0, 1\right]

— Represents the closed unit interval.

►ClosedInterval(1, 1) —

\left[1, 1\right]

— Represents the singleton set {1}.

►ClosedInterval(Neg(Infinity), 0) —

\left[-\infty, 0\right]

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

►ClosedInterval(1, -1) —

\left[1, -1\right]

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

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

1 + \left[0, 1\right] i

— Represents a set of points in the complex plane. ClosedInterval(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(ClosedInterval(1, 4), Mul(ClosedInterval(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
`ClosedInterval`	$\left[a, b\right]$	Closed interval
`RR`	$\mathbb{R}$	Real numbers
`Infinity`	$\infty$	Positive infinity
`ConstI`	$i$	Imaginary unit

Source code for this entry:

Entry(ID("12d5ab"),
    SymbolDefinition(ClosedInterval, ClosedInterval(a, b), "Closed interval"),
    CodeExample(ClosedInterval(a, b), "Represents", Set(x, ForElement(x, Union(RR, Set(Neg(Infinity), Infinity))), LessEqual(a, x, b)), "."),
    CodeExample(ClosedInterval(0, 1), "Represents the closed unit interval."),
    CodeExample(ClosedInterval(1, 1), "Represents the singleton set", Set(1), "."),
    CodeExample(ClosedInterval(Neg(Infinity), 0), "Represents half the extended real line (including minus infinity and zero)."),
    CodeExample(ClosedInterval(1, -1), "Represents the empty set.", " Note: potentially confusing rendering."),
    CodeExample(Add(1, Mul(ClosedInterval(0, 1), ConstI)), "Represents a set of points in the complex plane. ", SourceForm(ClosedInterval(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(ClosedInterval(1, 4), Mul(ClosedInterval(0, 1), ConstI)), "Represents a rectangle in the complex plane. "))

3fe68f

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. "))

b2162a

Symbol: ClosedOpenInterval —

\left[a, b\right)

— Closed-open interval

►ClosedOpenInterval(a, b) —

\left[a, b\right)

— Represents

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

►ClosedOpenInterval(0, 1) —

\left[0, 1\right)

— Represents the unit interval (including 0, excluding 1).

►ClosedOpenInterval(1, 1) —

\left[1, 1\right)

— Represents the empty set.

►ClosedOpenInterval(Neg(Infinity), 0) —

\left[-\infty, 0\right)

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

►ClosedOpenInterval(1, -1) —

\left[1, -1\right)

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

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

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

— Represents a set of points in the complex plane. ClosedOpenInterval(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(ClosedOpenInterval(1, 4), Mul(ClosedOpenInterval(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
`ClosedOpenInterval`	$\left[a, b\right)$	Closed-open interval
`RR`	$\mathbb{R}$	Real numbers
`Infinity`	$\infty$	Positive infinity
`ConstI`	$i$	Imaginary unit

Source code for this entry:

Entry(ID("b2162a"),
    SymbolDefinition(ClosedOpenInterval, ClosedOpenInterval(a, b), "Closed-open interval"),
    CodeExample(ClosedOpenInterval(a, b), "Represents", Set(x, ForElement(x, Union(RR, Set(Neg(Infinity), Infinity))), And(LessEqual(a, x), Less(x, b))), "."),
    CodeExample(ClosedOpenInterval(0, 1), "Represents the unit interval (including 0, excluding 1)."),
    CodeExample(ClosedOpenInterval(1, 1), "Represents the empty set."),
    CodeExample(ClosedOpenInterval(Neg(Infinity), 0), "Represents half the extended real line (including minus infinity, excluding zero)."),
    CodeExample(ClosedOpenInterval(1, -1), "Represents the empty set.", " Note: potentially confusing rendering."),
    CodeExample(Add(1, Mul(ClosedOpenInterval(0, 1), ConstI)), "Represents a set of points in the complex plane. ", SourceForm(ClosedOpenInterval(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(ClosedOpenInterval(1, 4), Mul(ClosedOpenInterval(0, 1), ConstI)), "Represents a rectangle in the complex plane. "))

ed302a

Symbol: OpenClosedInterval —

\left(a, b\right]

— Open-closed interval

►OpenClosedInterval(a, b) —

\left(a, b\right]

— Represents

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

►OpenClosedInterval(0, 1) —

\left(0, 1\right]

— Represents the unit interval (excluding 0, including 1).

►OpenClosedInterval(1, 1) —

\left(1, 1\right]

— Represents the empty set.

►OpenClosedInterval(Neg(Infinity), 0) —

\left(-\infty, 0\right]

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

►OpenClosedInterval(1, -1) —

\left(1, -1\right]

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

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

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

— Represents a set of points in the complex plane. OpenClosedInterval(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(OpenClosedInterval(1, 4), Mul(OpenClosedInterval(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
`OpenClosedInterval`	$\left(a, b\right]$	Open-closed interval
`RR`	$\mathbb{R}$	Real numbers
`Infinity`	$\infty$	Positive infinity
`ConstI`	$i$	Imaginary unit

Source code for this entry:

Entry(ID("ed302a"),
    SymbolDefinition(OpenClosedInterval, OpenClosedInterval(a, b), "Open-closed interval"),
    CodeExample(OpenClosedInterval(a, b), "Represents", Set(x, ForElement(x, Union(RR, Set(Neg(Infinity), Infinity))), And(Less(a, x), LessEqual(x, b))), "."),
    CodeExample(OpenClosedInterval(0, 1), "Represents the unit interval (excluding 0, including 1)."),
    CodeExample(OpenClosedInterval(1, 1), "Represents the empty set."),
    CodeExample(OpenClosedInterval(Neg(Infinity), 0), "Represents half the extended real line (excluding minus infinity, including zero)."),
    CodeExample(OpenClosedInterval(1, -1), "Represents the empty set.", " Note: potentially confusing rendering."),
    CodeExample(Add(1, Mul(OpenClosedInterval(0, 1), ConstI)), "Represents a set of points in the complex plane. ", SourceForm(OpenClosedInterval(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(OpenClosedInterval(1, 4), Mul(OpenClosedInterval(0, 1), ConstI)), "Represents a rectangle in the complex plane. "))