Definition:Real Interval/Unbounded Closed
Definition
There are two unbounded closed intervals involving a real number $a \in \R$, defined as:
\(\ds \hointr a \to\) | \(:=\) | \(\ds \set {x \in \R: a \le x}\) | ||||||||||||
\(\ds \hointl \gets a\) | \(:=\) | \(\ds \set {x \in \R: x \le a}\) |
Notation
An arbitrary (real) interval is frequently denoted $\mathbb I$.
Sources which use the $\textbf {boldface}$ font for the number sets $\N, \Z, \Q, \R, \C$ tend also to use $\mathbf I$ for this entity.
Some sources merely use the ordinary $\textit {italic}$ font $I$.
Some sources prefer to use $J$.
Wirth Interval Notation
The notation used on this site to denote a real interval is a fairly recent innovation, and was introduced by Niklaus Emil Wirth:
\(\ds \openint a b\) | \(:=\) | \(\ds \set {x \in \R: a < x < b}\) | Open Real Interval | |||||||||||
\(\ds \hointr a b\) | \(:=\) | \(\ds \set {x \in \R: a \le x < b}\) | Half-Open (to the right) Real Interval | |||||||||||
\(\ds \hointl a b\) | \(:=\) | \(\ds \set {x \in \R: a < x \le b}\) | Half-Open (to the left) Real Interval | |||||||||||
\(\ds \closedint a b\) | \(:=\) | \(\ds \set {x \in \R: a \le x \le b}\) | Closed Real Interval |
The term Wirth interval notation has consequently been coined by $\mathsf{Pr} \infty \mathsf{fWiki}$.
Also denoted as
The notation using $\infty$ is usual:
\(\ds \hointr a \infty\) | \(:=\) | \(\ds \set {x \in \R: a \le x}\) | ||||||||||||
\(\ds \hointl {-\infty} a\) | \(:=\) | \(\ds \set {x \in \R: x \le a}\) |
On $\mathsf{Pr} \infty \mathsf{fWiki}$ the $\gets \cdots \to$ notation is preferred.
Also known as
Some sources refer to the unbounded closed real intervals as half-open infinite (real) intervals.
Some sources use the term semi-infinite (closed) intervals.
Examples
Example $1$
Let $I$ be the unbounded closed real interval defined as:
- $I := \hointl \gets 3$
Then $2 \in I$.
Also see
- Definition:Open Real Interval
- Definition:Closed Real Interval
- Definition:Half-Open Real Interval
- Definition:Unbounded Open Real Interval
- Definition:Unbounded Real Interval without Endpoints
Sources
- 1970: Arne Broman: Introduction to Partial Differential Equations ... (previous) ... (next): Chapter $1$: Fourier Series: $1.1$ Basic Concepts: $1.1.1$ Definitions
- 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 3$. Ordered pairs; cartesian product sets (in passing)
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): Notation and Terminology: $\text{(v)}, \ \text{(vii)}$
- 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 2$: Continuum Property: $\S 2.9$: Intervals
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 6$: Subsets
- 1991: Felix Hausdorff: Set Theory (4th ed.) ... (previous) ... (next): Preliminary Remarks
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): interval: $\text {(v)}$, $\text {(vii)}$