Definition:Real Interval/Half-Open/Left
Definition
Let $a, b \in \R$ be real numbers.
The left half-open (real) interval from $a$ to $b$ is the subset:
- $\hointl a b := \set {x \in \R: a < x \le b}$
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 defined as
Some sources, when defining a half-open real interval, require that $a < b$.
This is to eliminate the degenerate case where the interval is the empty set.
Also known as
A left half-open interval is also called:
Also see
Technical Note
The $\LaTeX$ code for \(\hointl {a} {b}\) is \hointl {a} {b}
.
This is a custom $\mathsf{Pr} \infty \mathsf{fWiki}$ command designed to implement Wirth interval notation.
The name is derived from half-open interval on the left.
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
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): Notation and Terminology: $\text{(iii)}$
- 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
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.2$: Numbers, Powers, and Logarithms
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): closed interval
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): interval
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): open interval
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): closed interval
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): interval
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): open interval
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): interval: $\text {(iv)}$