# Numeric Interval

(Redirected from Closed Interval)

Jump to navigation
Jump to search
A numeric interval is a contiguous numeric subsequence of a formal number sequence.

**AKA:**Interval, Number Interval, Range.**Context:**- It can have a Minimum Number (Infimum)
*i*. - It can have a Maximum Number (Supremum)
*s*. - It can be:
- an Empty Interval, with Zero Members.
- a Degenerate Interval, with One Member.
- a Proper Interval, otherwise.

- It can be:
- a Finite Interval, e.g. [0,1,2,3,4]
- a Countable Interval, e.g. [math]\displaystyle{ [0,1,2,3,...,\infty] }[/math]
- an Uncountable Interval, e.g. [math]\displaystyle{ (0,...,\pi/4,...,\infty) }[/math]

- It can be
- a Bounded Interval, in which neither the Infimum or Supremum are the Infinite Number:
- an Open Interval: [math]\displaystyle{ (i,s)=\{x \vert i\lt x\lt s\} }[/math]
- an Closed Interval: [
*i,s*]={*x*|*i*≤*x*≤*s*} - a Left-Closed Right-Open Interval: [
*i*,*s*)={*x*|*i*≤*x*<s*}* - a Left-Open Right-Closed Interval: (
*i*,*s*]={*x*|*i*<*x*≤*s*}

- a Partially Bound Interval, in which one of the Infimum or Supremum is the Infinite Number:
- a Right-Open Interval: [math]\displaystyle{ (-\infty,s)=\{x \vert x \lt s\} }[/math]
- a Right-Closed Interval: [math]\displaystyle{ (-\infty,s]=\{x \vert x \le s\} }[/math]

- a Bounded Interval, in which neither the Infimum or Supremum are the Infinite Number:
- It can be:
- It can be an input to a Length Function(Diameter Function / Width Function)

- It can have a Minimum Number (Infimum)
**Example(s):**- [1,0] ⇒ {}, an Empty Interval.
- [1,1] ⇒ {1}, a Degenerate Interval.
- a Finite Interval.
- [1,2]
_{I}⇒ {1 < 2} - [1,100]
_{I}⇒ {1 < 2 < 3 … < 100} a Finite Integer Interval, for The Integer Number Sequence.

- [1,2]
- a Countable Interval.
- an Uncountable Interval.
- [0,1)
_{R}⇒ {0, ..., 0.999999...} a Real Number Interval, from The Real Number Sequence. - A Time Interval, with a Start Time and an End Time.

- [0,1)
- …

**Counter-Example(s):**- {1 < 3 < 5}, a Number Sequence.
- The Prime Number Sequence.
- The Fibonacci Number Sequence.
- ?? (-∞, ∞)
_{R}, The Real Number Sequence.

**See:**Unit Function, Partially Ordered Set, Range, Inequality Relation, Interval Scale.

## References

### 2011

- (Wikipedia, 2011) ⇒ http://en.wikipedia.org/wiki/Interval_(mathematics)
- …
**Terminology**- An
**open interval**does not include its endpoints, and is indicated with parentheses. For example (0,1) means greater than 0 and less than 1. Conversely, a**closed interval**includes its endpoints, and is denoted with square brackets. For example [0,1] means greater than or equal to 0 and less than or equal to 1. - A
**degenerate interval**is any set consisting of a single real number. Some authors include the empty set in this definition. An interval that is neither empty nor degenerate is said to be proper, and has infinitely many elements. - An interval is said to be
**left-bounded**or right-bounded if there is some real number that is, respectively, smaller than or larger than all its elements. An interval is said to be**bounded**if it is both left- and right-bounded; and is said to be unbounded otherwise. Intervals that are bounded at only one end are said to be**half-bounded**. The empty set is bounded, and the set of all reals is the only interval that is unbounded at both ends. Bounded intervals are also commonly known as finite intervals. - Bounded intervals are bounded sets, in the sense that their diameter (which is equal to the absolute difference between the endpoints) is finite. The diameter may be called the
**length**, width,**measure**, or size of the interval. The size of unbounded intervals is usually defined as +∞, and the size of the empty interval may be defined as 0 or left undefined. - The
**centre**of bounded interval with endpoints`a`and`b`is (*a*+*b*)/2, and its**radius**is the half-length |*a*−*b*|/2. These concepts are undefined for empty or unbounded intervals. - An interval is said to be
**left-open**if and only if it has no minimum (an element that is smaller than all other elements); right-open if it has no maximum; and**open**if it has both properties. The interval [0,1) = {*x*| 0 ≤*x*< 1}, for example, is left-closed and right-open. The empty set and the set of all reals are open intervals, while the set of non-negative reals, for example, is a right-open but not left-open interval. The open intervals coincide with the open sets of the real line in its standard topology. - An interval is said to be
**left-closed**if it has a minimum element, right-closed if it has a maximum, and simply**closed**if it has both. These definitions are usually extended to include the empty set and to the (left- or right-) unbounded intervals, so that the closed intervals coincide with closed sets in that topology. - The
**interior**of an interval`I`is the largest open interval that is contained in`I`; it is also the set of points in`I`which are not endpoints of`I`. The**closure**of`I`is the smallest closed interval that contains`I`; which is also the set`I`augmented with its finite endpoints. - For any set
`X`of real numbers, the**interval enclosure**or interval span of`X`is the unique interval that contains`X`and does not properly contain any other interval that also contains`X`.

- An

- …