half-open interval
C2Technical/Formal
Definition
Meaning
A set of real numbers that includes one endpoint but not the other, typically written as [a, b) or (a, b], meaning all numbers x such that a ≤ x < b or a < x ≤ b, respectively.
In mathematics, a specific type of interval that is neither fully closed (including both endpoints) nor fully open (excluding both endpoints). It can represent ranges in domains like time, measurement, or probability where one boundary is inclusive and the other is exclusive.
Linguistics
Semantic Notes
The term is almost exclusively used in mathematical, statistical, and computational contexts. The concept is precise and avoids ambiguity in defining boundaries. The notation is standardized, with the square bracket denoting inclusion and the parenthesis denoting exclusion.
Dialectal Variation
British vs American Usage
Differences
No significant differences in meaning or usage. Terminology and notation are identical in British and American mathematical English.
Connotations
None beyond its technical precision.
Frequency
Equally low-frequency and specialized in both varieties, confined to technical fields.
Vocabulary
Collocations
Grammar
Valency Patterns
the half-open interval [from] X [to] Ya half-open interval [on] the real lineVocabulary
Synonyms
Neutral
Weak
Vocabulary
Antonyms
Usage
Context Usage
Business
Virtually never used.
Academic
Used in mathematics, computer science (e.g., for array indexing), statistics, and engineering courses and literature.
Everyday
Not used in everyday conversation.
Technical
The primary context. Used to precisely define ranges in mathematical proofs, algorithm specifications, and scientific modelling.
Examples
By Part of Speech
adjective
British English
- The function is defined on a half-open interval.
American English
- We need a half-open interval for the integral's domain.
Examples
By CEFR Level
- In mathematics, a half-open interval includes one endpoint but not the other.
- The data is valid for the half-open interval [start_time, end_time), meaning the end time itself is excluded from the set.
- Array indices in many programming languages naturally correspond to a half-open interval, which helps avoid off-by-one errors.
Learning
Memory Aids
Mnemonic
Think of a door that is AJAR — it's neither fully open nor fully closed. A half-open interval is like that door for numbers: one end is 'shut' (inclusive), the other is 'open' (exclusive).
Conceptual Metaphor
A RANGE WITH A SOFT AND A HARD BOUNDARY: One side is a firm wall you can touch (inclusive), the other is a line you can approach but not cross (exclusive).
Watch out
Common Pitfalls
Translation Traps (for Russian speakers)
- Avoid literal translation 'полуоткрытый интервал' unless in a strict mathematical context; it is not a general phrase.
- Do not confuse with 'полуинтервал', which is a direct equivalent but may be less common in some Russian pedagogical traditions.
Common Mistakes
- Writing the notation incorrectly, e.g., mixing up [a, b) with (a, b].
- Pronouncing it as 'half-opened interval'.
- Using it in non-mathematical contexts where 'range' or 'period' would be appropriate.
Practice
Quiz
What does the notation (3, 7] represent?
FAQ
Frequently Asked Questions
Yes, 'semi-open interval' is a less common but perfectly acceptable synonym for 'half-open interval'.
They are useful because they eliminate ambiguity when dealing with consecutive ranges. For example, splitting a range [0, N) into [0, M) and [M, N) leaves no gap and no overlap.
Yes. For example, the half-open interval [5, 5) is empty because it includes numbers x where 5 ≤ x < 5, which is impossible. It contains no points.
It is typically read aloud as 'the interval from a to b, including a but excluding b' or 'the closed-open interval from a to b'. The bracket shapes are described verbally to specify inclusivity.