Numeric Interval

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]
  • It can be:
    • an Integer Number Interval.
    • a Real Number Interval.
  • It can be an input to a Length Function(Diameter Function / Width Function)
• 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.
  • a Countable Interval.
    • a Rational Number Interval, from The Rational Number Sequence.
  • 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.
  • …
• 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.

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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.

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