Interval (mathematics)

Intervals are called sets of real numbers whose elements x are between two numbers a and b , which satisfy the condition a ≤ b . These are real numbers or represent positive or negative growth trend: symbolized by + ∞ or -∞

Intuitively we imagine a thin and stretched thread, we consider two points where we cut the thread: the piece we obtain is the image of a finite interval, the cut points may or may not be in the piece. In the case of taking only one point and cutting, any of the separated pieces is the image of an infinite interval, the same as the whole thread.

Summary

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  • 1 Components
  • 2 Classification
    • 1 For the belonging of its ends
    • 2 By length
    • 3 Notations
  • 3 Cases
  • 4 Joint operations
  • 5 Uses
  • 6 Sources

Components

  • Ends: a and b
  • Interior formed by points x such that a <x <b
  • Length is ba
  • Midpoint m = (a + b) ÷ 2

Classification

For the belonging of its ends

  • open
  • closed
  • semi-open

By length

  • Finite
  • Infinite

Notations

  • For closed intervals [] is used
  • For open intervals (), <>,] [
  • Half open combining the two previous cases

Cases

  1. [a, b] closed interval of finite length l= b – a. a ≤ x ≤ b .
  2. [a, b [or [a, b) interval closed at a, open at b (semi-closed, semi-open), of finite length l= b – a. a ≤ x <b .
  3. ] a, b] or (a, b] interval open at a, closed at b, of finite length l= b – a. a <x ≤ b .
  4. ] a, b [or (a, b) open interval, of finite length l= b – a. a <x <b .
  5. ] – ∞, b [or (- ∞, b) open interval of infinite length. x <b.
  6. ] – ∞, b] or (- ∞, b] (semi) closed interval of infinite length x ≤ b.
  7. [a, + ∞ [(semi) closed interval of infinite length. a ≤ x.
  8. ] a, + ∞ [or (a, + ∞) open interval of infinite length. a <x.
  9. ] – ∞, + ∞ [o (- ∞, + ∞) or R, interval both open and closed, of infinite length. R x belongs to .
  10. {a} closed interval of null length. It is a unitary set. (corresponds to case a = b). x = a
  11. {} = ∅ the empty set, interval both open and closed. x does not exist.

Joint operations

  • Union, intersection, difference, and symmetric difference of sets can be performed.
  • The results are not necessarily sets: <2, 5> junction <7.10> exists but is not interval, the same [1, 10] – [3, 7] is the union of two semi-open intervals. [1,> junction <7.10]

Applications

  • In differential calculus and mathematical analysis, the open interval containing the accumulation point, limit case, is used; or the point where the derivative is defined.
  • Any type of interval to define or find the domain and codomain of a real function of real variable.
  • In integral calculus to find the domain of a definite integral, or the limits of integration
  • In usual line topology the open interval is used to define neighborhood.
  • In another topology the semi-open interval.

 

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