Definition:Closed Rectangle
Definition
Let $n \ge 1$ be a natural number.
Let $a_1, \ldots, a_n, b_1, \ldots, b_n$ be real numbers.
The Cartesian product of closed intervals:
- $\ds \prod_{i \mathop = 1}^n \closedint {a_i} {b_i} = \closedint {a_1} {b_1} \times \cdots \times \closedint {a_n} {b_n} \subseteq \R^n$
is called a closed rectangle in $\R^n$ or closed $n$-rectangle.
Degenerate Case
In the case where $a_i > b_i$ for some $i$, the closed rectangle $\ds \prod_{i \mathop = 1}^n \closedint {a_i} {b_i}$ degenerates to the empty set $\O$.
This is in accordance with the result Cartesian Product is Empty iff Factor is Empty for general Cartesian products.
Notation
A convenient notation for the closed rectangle $\ds \prod_{i \mathop = 1}^n \closedint {a_i} {b_i}$ is $\closedrect {\mathbf a} {\mathbf b}$.
Also see
- Results about closed rectangles can be found here.
Technical Note
The $\LaTeX$ code for \(\closedrect {\mathbf a} {\mathbf b}\) is \closedrect {\mathbf a} {\mathbf b} .
This is a custom $\mathsf{Pr} \infty \mathsf{fWiki}$ command designed to implement Wirth interval notation and its derivatives.
Sources
- 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
- 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