## Intervals

Table of Contents

## 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.

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:

Equation | Details |

x = y | x is equal to y |

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

Symbol | Rule |

> | 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 anopen sentence.

### Interval Notation

Symbol | Rule |

[ ] | 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

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

To include 0, and not include 7:

Inequalities | 0 ≤ x < 7 |

Number line | |

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 Notation | Inequalities | Details |

(a, b) | a < x < b | an open interval |

[a, b) | a ≤ x < b | closed on left, open on right |

(a, b] | a < x ≤ b | open on left, closed on right |

[a, b] | a ≤ x ≤ b | a 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 Notation | Inequalities | Details |

(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 |

(-∞, ∞) | –∞ < 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.

Relation | Symbol | Definition |

Union | A ⋃ B | the set of elements which are in either set |

Intersection | A ⋂ B | the 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”.

Inequalities | Interval 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”.

Inequalities | Interval Notation | Intersection |

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] |