In mathematics, specifically within the context of interval notation, "backwards brackets" (also known as reversed brackets) are a notation used to indicate that an endpoint of an interval is exclusive, meaning the number at that endpoint is not included in the interval.
Understanding Reversed Brackets in Interval Notation
As noted by many authors, using reversed brackets like ] or [ instead of standard parentheses ( or ) is an alternative way to denote an open interval or the exclusive end of a half-open or half-closed interval. This notation explicitly points away from the numbers outside the interval, rather than enclosing the numbers inside.
Based on the reference provided:
- The notation ]5,7[ refers to the interval from 5 to 7, exclusive. This means the interval includes all numbers between 5 and 7, but it does not include 5 or 7 themselves.
- Similarly, the example interval from -1 to 2, exclusive of 2, would be written as [–1, 2[. This indicates all numbers from -1 up to (but not including) 2. The standard bracket [ indicates that -1 is included in the interval.
This reversed bracket notation serves the same purpose as using parentheses ( and ).
Comparing Interval Notation Styles
Here's a quick comparison of common notations for intervals:
| Type of Interval | Description | Common Notation | Reversed Bracket Notation | Example (Exclusive) | Example (Inclusive/Half) |
|---|---|---|---|---|---|
| Open Interval | Includes numbers between endpoints, not endpoints themselves (exclusive). | (a, b) |
]a, b[ |
(5, 7) or ]5, 7[ |
N/A |
| Closed Interval | Includes numbers between and including endpoints (inclusive). | [a, b] |
[a, b] |
N/A | [-1, 2] |
| Half-Open / Half-Closed | Includes one endpoint but not the other. | [a, b) or (a, b] |
[a, b[ or ]a, b] |
[-1, 2) or [-1, 2[ |
(-1, 2] or ] -1, 2] |
Why Use Reversed Brackets?
While parentheses are widely used and perfectly clear, some mathematicians prefer reversed brackets for certain contexts, perhaps to visually emphasize that the boundary is not part of the set. It's essentially a stylistic or historical alternative to the more common parentheses for denoting exclusive boundaries.
Understanding this notation is important when encountering mathematical texts or papers that adopt this style, as it directly impacts the meaning of the interval being described.
