Intervals

Definition

In mathematics, interval is a set of real numbers with the property that any number that lies between 2 numbers in the set is also included in the set.

For example, the set of all numbers x satisfying 3 ≤ x ≤ 5 is an interval which contains 3, 5, and all numbers between them.

Interval: all numbers between two given numbers.

Examples

Interval 3 to 5 includes:

3.1

3.1111

3.5

4.75

4.80001

9/2

4.9937

Intervals are central to interval arithmetic, a general numerical computing technique that automatically provides guaranteed enclosures for arbitrary formulas, even in the presence of uncertainties, arithmetic roundoff, & mathematical approximations.

Real intervals play an important role in the theory of integration. They are the simplest sets whose size, measure, or length is easy to define.

The concept of measure can then be extended to more complicated sets of real numbers.

Inequalities, Interval Notation, & Number line

Inequalities

Equations and inequalities are mathematical sentences formed by relating 2 expressions to each other. In an equation, the 2 expressions are deemed equal which is shown by the symbol =.

Example:

EquationDetails
x = yx is equal to y

Where as in an inequality, the two expressions are not necessarily equal which is indicated by the symbols: >, <, ≤ or ≥.

SymbolRule
>greater than
greater than or equal to
<less than
less than or equal to

An equation or an inequality that contains at least 1 variable is called an open sentence.

Interval Notation

SymbolRule
[ ]include the end value
( )exclude the end value

The interval of numbers between a and b, including a and b, is often denoted [a, b]. The 2 numbers are called the endpoints of the interval.

Number line

Number Line

all the numbers between 4 and 15, do not include 4, but do include 15

Examples

To include 0, and not include 7:

Inequalities0 ≤ x < 7
Number lineNumber Line Example
Interval Notation [0, 7)

Open or Closed

To indicate that one of the endpoints is to be excluded from the set, the corresponding square bracket can be either replaced with a parenthesis, or reversed. Thus, in set builder notation:

Interval NotationInequalitiesDetails
(a, b)a < x < ban open interval
[a, b)a ≤ x < bclosed on left, open on right
(a, b]a < x ≤ bopen on left, closed on right
[a, b]a ≤ x ≤ ba closed interval

Note:

(a, a), [a, a), and (a, a] each represents the empty set, whereas [a, a] denotes the set {a}. When a > b, all 4 notations are usually taken to represent the empty set.

To Infinity (Not Beyond):

an interval may be defined as a subset of the extended real numbers, the set of all real numbers augmented with +∞ and −∞.

Interval NotationInequalitiesDetails
(a, +∞)x > agreater than a
[a, +∞)x ≥ agreater than or equal to a
(-∞, a)x < aless than a
(-∞, a]x ≤ aless than or equal to a
(-∞, ) < x < ∞no limit

In this interpretation, the notations above are all meaningful and distinct. In particular, (−∞, +∞) denotes the set of all ordinary real numbers, while [−∞, +∞] denotes the extended reals.

Union and Intersection

We can have 2 (or more) intervals.

Given 2 intervals, the two major relations between them are their union and their intersection.

RelationSymbolDefinition
UnionA Bthe set of elements which are in either set
IntersectionA Bthe set of elements which are in both set

Union

The union of 2 sets is a new set that contains all of the elements that are in at least 1 of the 2 sets. The union is written as A∪B or “A or B”.

InequalitiesInterval Notation
x < 4 or x > 7(-∞, 4) ⋃(7, +∞)
x ≤ 2 or x > 5(-∞, 2] ⋃(5, +∞)
x < 1 or x ≥ 8(-∞, 1) ⋃[8, +∞)
x ≤ 0 or x ≥ 9(-∞, 0] ⋃[9, +∞)

Intersection

The intersection of 1 sets is a new set that contains all of the elements that are in both sets. The intersection is written as A∩B or “A and B”.

InequalitiesInterval NotationIntersection
x < 7 and x > 4(-∞, 7) ⋂ (4, +∞)(4, 7)
x ≤ 5 and x > 2(-∞, 5] ⋂ (2, +∞)(2, 5]
x < 8 and x ≥ 1(-∞, 8) ⋂ [1, +∞)[1, 8)
x ≤ 9 and x ≥ 0(-∞, 9] ⋂ [0, +∞)[0, 9]

Learn More

Injective, Surjective & Bijective

Set-Builder Notation

Function Transformations

Inverse Functions

Set Index

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