interval

interval

Loosely speaking, an

interval

is a part of the

real numbers

that start at one number and stops at another number. For instance,
all numbers

greater

that

1

and smaller than

2

form in interval.
Another interval is formed by numbers greater or equal to

1

and
smaller than

2

. Thus, when talking about intervals, it is necessary
to specify whether the endpoints are part of the interval or not.
There are then four types of intervals with three different names:
open, closed and half-open.
Let us next define these precisely.

1.

The open interval contains neither of the endpoints.
If

a

and

b

are real numbers, then the

open interval

of
numbers between

a

and

b

is written as

(

a

,

b

)

and

(

a

,

b

)

=

{

x

∈

ℝ

∣

a

<

x

<

b

}

.

2.

The

closed interval

contains both endpoints.
If

a

and

b

are real numbers, then the

closed interval

is written as

[

a

,

b

]

and

[

a

,

b

]

=

{

x

∈

ℝ

∣

a

≤

x

≤

b

}

.

3.

A half-open interval contains only one of
the endpoints.
If

a

and

b

are real numbers, the

half-open intervals

(

a

,

b

]

and

[

a

,

b

)

are defined as

(

a

,

b

]

=

{

x

∈

ℝ

∣

a

<

x

≤

b

}

,

⁢

[

a

,

b

)

=

{

x

∈

ℝ

∣

a

≤

x

<

b

}

.

Note that this definition includes the empty set as an interval by, for example, taking the interval

(

a

,

a

)

for any

a

.

An interval is a subset

S

of a

totally ordered set

T

with the property that whenever

x

and

y

are in

S

and

x

<

z

<

y

then

z

is in

S

. Applied to the real numbers, this encompasses open, closed, half-open, half-infinite,

infinite

, empty, and one-point intervals. All the various different types of interval in

ℝ

have this in common. Intervals in

ℝ

are

connected

under the usual topology.

There is a standard way of graphically representing intervals
on the real line using filled and empty circles. This is illustrated in
the below figures:

The logic is here that a empty circle represent a point not belonging
to the interval, while a filled circle represents a point belonging
to the interval. For example, the first interval is an open interval.

Infinite intervals

If we allow either (or both) of

a

and

b

to be infinite, then we
define

(

a

,

∞

)

=

{

x

∈

ℝ

∣

x

>

a

}

,

⁢

[

a

,

∞

)

=

{

x

∈

ℝ

∣

x

≥

a

}

,

(

-

∞

,

a

)

=

{

x

∈

ℝ

∣

x

<

a

}

,

(

-

∞

,

a

]

=

{

x

∈

ℝ

∣

x

≤

a

}

,

(

-

∞

,

∞

)

=

ℝ

.

The graphical representation of infinite intervals is as follows:

Note on naming and notation

In

[

1

,

2

]

, an open interval is always called
a

segment

, and a closed interval is called simply an interval.
However, the above naming with open, closed, and half-open interval seems
to be more widely adopted.
See e.g.

[

3

,

4

,

5

]

. To distinguish between

[

a

,

b

)

and

(

a

,

b

]

, the former is sometimes called a

right half-open interval

and
the latter a

left half-open interval

[

6

]

.
The notation

(

a

,

b

)

,

[

a

,

b

)

,

(

a

,

b

]

,

[

a

,

b

]

seems to be standard. However,
some authors (especially from the French school) use notation

]

a

,

b

[

,

[

a

,

b

[

,

]

a

,

b

]

,

[

a

,

b

]

instead of the above (in the same ).

Bourbaki

, for example, uses this notation.

This entry contains content adapted from the

Wikipedia

article

http://en.wikipedia.org/wiki/Interval_(mathematics)

Interval (mathematics) as of November 10, 2006.

References

1

W. Rudin,

Principles of

Mathematical Analysis

, McGraw-Hill Inc., 1976.

2

W. Rudin,

Real and

complex analysis

, 3rd ed., McGraw-Hill Inc., 1987.

3

R. Adams,

Calculus

, a

complete

course

,
Addison-Wesley Publishers Ltd., 3rd ed., 1995.

4

L. Råde, B. Westergren,

Mathematics Handbook for Science and Engineering

,
Studentlitteratur, 1995.

5

R.A. Silverman,

Introductory Complex Analysis

,
Dover Publications, 1972.

6

S. Igari,

Real analysis

- With an introduction to Wavelet Theory

,

American Mathematical Society

, 1998.

The metapost code for the figures can be found

http://aux.

planetmath

.org/files/objects/4446/

here.

Title

interval

Canonical name

Interval

Date of creation

2013-03-22 13:44:58

Last modified on

2013-03-22 13:44:58

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

16

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 12D99

Classification

msc 26-00

Classification

msc 54C30

Related topic

OpenSetsInMathbbRnContainsAnOpenRectangle

Related topic

LineSegment

Related topic

CircularSegment

Defines

open interval

Defines

closed interval

Defines

half-open interval

Defines

right half-open interval

Defines

left-half-open interval

Defines

segment

Generated on Fri Feb 9 16:39:45 2018 by

LaTeXML