Interval is a set of real numbers that contains all real numbers lying between any two numbers of the set

  • Real Number

  • Numbers that include both rational and irrational numbers:

    Whole numbers such as (0, 1, 2, 3, …)

    Rational numbers such as (1/2, 2.5, 0.123, etc) and

    Irrational numbers such as √3, π (22/7), etc

  • What is not a Real number

  • Imaginary number such as π-1 ,

    Infinity is not a real number

  • Type of intervals

  • Open Intervals:

    Contains each value between the end points but does not include the end points. Example, open interval (2, 5) contains each real number lying between 2 and 5 but does not contain 2 and 5.

    Closed Intervals

    Contains each value between and including extreme values. Example, closed interval [2, 5] contains each real number lying between 2 and 5. It also contains its end points. i

    Left Open and Right Closed interval

    Left Closed and Right Open interval

    Interval Notation

    we just write the beginning and ending numbers of the interval, and use

    [ ] a square bracket when we want to include the end value, or ( ) a round bracket when we don't

Tabular Presentation

Notation Inequality Description Eample
(a,b) a < x < b An open interval (1,10): neither one included nor 10 included in Set {2, 3, 4, 5, 6, 7, 8, 9}
[a, b) a ≤ x < b closed on left, open on right [1,10): one included but 10 not included in Set {1, 2, 3, 4, 5, 6, 7, 8, 9}
(a, b] a < x ≤ b open on left, closed on right (1,10]: one not included but 10 included in Set {2, 3, 4, 5, 6, 7, 8, 9, 10}
[a, b] a ≤ x ≤ b a closed interval [1,10]: one and 10 both end included in Set {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
  • Other Type of intervals

  • Finite Intervals:

    An interval is said to be finite if its length is finite. For example, if I = (– 3, 5)

    Infinite Intervals

    An interval is said to be infinite interval if its length is not finite.
    Example, (i) The set {x ∈ R : x > a, a ∈ R}is an infinite interval and is denoted by (a, ∞

    There are 4 possible "infinite ends"

    Interval Inequality Description
    (a, + ∈) x > a "greater than a"
    [a, + ∈) x ≥ a "greater than or equal to a"
    (- ∈, a) x < a "less than a"
    (- ∈, a] x ≤ a "less than or equal to a"
    We could even show no limits by using this notation: (- ∈, + ∈)