Interval (mathematics)

An interval in calculus, order topology and related fields of mathematics is a "connected" subset of a totally (or linearly) ordered carrier set (for example the set of real numbers $\mathbb {R}$ ). A (bounded) interval consists of all elements , which can be compared in size with two bounding elements of the carrier set, the lower bound and the upper bound the interval, and which lie between the boundaries in the sense of this comparison. Thereby the boundaries of the interval can belong to the interval (closed interval, $a\leq x\leq b$ ), not belong to it (open interval partially belong (half-open interval, a ≤ x < $a\leq x<b$ $a<x\leq b$ ).

Contiguous means here: If two objects are contained in the subset, then all objects that lie between them (in the carrier set) are also contained in it. The most important examples of carrier sets are the sets of real, rational, integer and natural numbers. In the above cases, and more generally whenever a difference is declared between two elements of the carrier set, the difference between the upper and lower bounds of the interval ( b-a ) is called the length of the interval, or interval length for short; the term interval diameter is also common for this difference. If an arithmetic mean of the interval boundaries is declared, this is called the interval center.

Examples

In the set of natural numbers

$\{5,6,7,8,9\}$

In this case of a discrete set, the elements of the interval are adjacent.

In the set of real numbers

$[0,1]=\{x\in \mathbb{R} \mid 0\leq x\leq 1\}$ ,

the set of all numbers between 0 and 1, where the endpoints 0 and 1 are included.

Trivial examples of intervals are the empty set and sets that have exactly one element. If one does not want to include these, then one speaks of real intervals.

The set $\{5,6,7,8,9\}$ can also be considered as a subset of the carrier set of real numbers. In this case, it is not an interval, since the set does not contain, for example, the non-natural numbers lying between 6 and 7.

The carrier set of the real numbers plays a special role among the mentioned carrier sets for intervals insofar as it is order-complete (see also Dedekind cut). Intervals are in this case exactly the subsets connected in the sense of the topology.

n-dimensional intervals

Definition

Analogously, for defines $n\in \mathbb {N}$ in the n-dimensional space $\mathbb {R} ^{n}$ any n-dimensional interval (cuboid).

$I:=I_{1}\times \dotsb \times I_{n}$ with arbitrary intervals $I_{1},\dotsc ,I_{n}\subseteq \mathbb {R} .$

Constrained n-dimensional intervals

Now let $a,b\in \mathbb{R} ^{n}$ with $a=(a_{1},\dotsc ,a_{n})$ and $b=(b_{1},\dotsc ,b_{n})$ , then specifically:

Completed interval

$[a,b]:=\{(x_{1},\dotsc ,x_{n})\in \mathbb {R} ^{n}\mid a_{1}\leq x_{1}\leq b_{1},\dotsc ,a_{n}\leq x_{n}\leq b_{n}\}.$

Open interval

$(a,b)={]{a,b}[}:=\{(x_{1},\dotsc ,x_{n})\in \mathbb {R} ^{n}\mid a_{1}<x_{1}<b_{1},\dotsc ,a_{n}<x_{n}<b_{n}\}.$

Half-open (more precisely right-open) interval

$[a,b)={[{a,b}[}:=\{(x_{1},\dotsc ,x_{n})\in \mathbb {R} ^{n}\mid a_{1}\leq x_{1}<b_{1},\dotsc ,a_{n}\leq x_{n}<b_{n}\}.$

Half-open (more precisely left-open) interval

$(a,b]={]{a,b}]}:=\{(x_{1},\dotsc ,x_{n})\in \mathbb {R} ^{n}\mid a_{1}<x_{1}\leq b_{1},\dotsc ,a_{n}<x_{n}\leq b_{n}\}.$

Generalization

In topology, real intervals are examples of connected sets; in fact, a subset of the real numbers is connected precisely if it is an interval. Open intervals are open sets and closed intervals are closed sets. Semi-open intervals are neither open nor closed. Closed bounded intervals are compact.

All notations made here for the real numbers $\mathbb {R}$ can be directly applied to any totally ordered set.

Questions and Answers

Q: What is an interval in mathematics?

A: An interval in mathematics is a group of numbers that includes all numbers between the beginning and the end.

Q: How do you determine which numbers are inside an interval?

A: Numbers that are larger than the beginning number and smaller than the end number are inside the interval, and numbers that are smaller than the beginning number or larger than the end number are not in the interval.

Q: Do both the beginning and end numbers have to be included in an interval?

A: The beginning number and end number may or may not be inside the interval.

Q: How do you write an interval?

A: To write an interval, write either a square bracket ( [ ) or a parenthesis ( ( ), then include the beginning number, followed by a comma ( , ), then include the end number, followed by either a closing square bracket ( ] ) or a closing parenthesis ( ).

Q: Can you give examples of intervals?

A: Examples of intervals are (4, 9.6), [-100, 100], [-30, -4).

Q: Are negative numbers allowed within an interval?

A: Yes, negative numbers can be included within an interval.