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Half-Open/Half-Closed Interval

Half-Closed Interval
Half-Open Interval

An interval that contains one endpoint but not the other.

 

Number line showing half-closed interval [-2, 3) with a filled dot at -2 and open dot at 3, connected by a line.

 

 

See also

Interval notation, closed interval, open interval

Key Formula

[a,b)={x∈R∣a≤x<b}or(a,b]={x∈R∣a<x≤b}[a, b) = \{x \in \mathbb{R} \mid a \leq x < b\} \qquad \text{or} \qquad (a, b] = \{x \in \mathbb{R} \mid a < x \leq b\}
Where:
  • aa = The left (lower) endpoint of the interval
  • bb = The right (upper) endpoint of the interval
  • xx = Any real number belonging to the interval
  • [[ = Square bracket means the endpoint is included (≤ or ≥)
  • (( = Parenthesis (or round bracket) means the endpoint is excluded (< or >)

Worked Example

Problem: Determine which numbers from the set {1, 2, 3, 4, 5, 6} belong to the half-open interval [2, 5).
Step 1: Write the interval in set-builder notation to clarify which values qualify.
[2,5)={x∈R∣2≤x<5}[2, 5) = \{x \in \mathbb{R} \mid 2 \leq x < 5\}
Step 2: Check the left endpoint. The square bracket at 2 means 2 is included. Since 2 ≤ 2 is true, 2 qualifies.
2≤2✓2 \leq 2 \quad \checkmark
Step 3: Check the right endpoint. The parenthesis at 5 means 5 is excluded. Since 5 < 5 is false, 5 does not qualify.
5<5×5 < 5 \quad \times
Step 4: Test the remaining values: 1 fails because 1 < 2. The values 3 and 4 satisfy 2 ≤ x < 5. The value 6 fails because 6 ≥ 5.
3∈[2,5),4∈[2,5),1∉[2,5),6∉[2,5)3 \in [2,5), \quad 4 \in [2,5), \quad 1 \notin [2,5), \quad 6 \notin [2,5)
Answer: The numbers from the set that belong to [2, 5) are 2, 3, and 4.

Another Example

This example uses the other form of half-open interval — open on the left, closed on the right — and asks the student to graph it, reinforcing the visual meaning of each bracket type.

Problem: Graph the half-open interval (−3, 1] on a number line and express it as an inequality.
Step 1: Identify the type. The parenthesis is on the left at −3 (excluded) and the square bracket is on the right at 1 (included).
(−3,1]={x∈R∣−3<x≤1}(−3, 1] = \{x \in \mathbb{R} \mid -3 < x \leq 1\}
Step 2: On the number line, draw an open circle at −3 to show it is not part of the interval.
Step 3: Draw a closed (filled-in) circle at 1 to show it is included in the interval.
Step 4: Shade the region between −3 and 1. As an inequality, the interval is:
−3<x≤1-3 < x \leq 1
Answer: The interval (−3, 1] is graphed with an open circle at −3, a closed circle at 1, and the segment between them shaded. The inequality form is −3 < x ≤ 1.

Frequently Asked Questions

What is the difference between a half-open interval and a half-closed interval?
There is no difference — they are two names for the same thing. Both terms describe an interval that includes exactly one endpoint and excludes the other. Some textbooks prefer 'half-open' while others use 'half-closed'; the notation [a, b) or (a, b] is the same either way.
How do you know which bracket to use: square or round?
A square bracket [ or ] means the endpoint is included in the interval (the inequality uses ≤ or ≥). A round bracket ( or ) means the endpoint is excluded (the inequality uses < or >). For example, [3, 7) includes 3 but excludes 7.
When would you use a half-open interval instead of a closed or open interval?
Half-open intervals arise naturally in many settings. For instance, when you partition a range into non-overlapping pieces — like age groups 0 ≤ age < 10 and 10 ≤ age < 20 — each piece is a half-open interval so that every value falls into exactly one group. They also appear in piecewise functions and computer science (array indexing).

Half-Open Interval vs. Closed Interval vs Open Interval

Half-Open IntervalClosed Interval vs Open Interval
Endpoints includedExactly one endpointClosed: both endpoints; Open: neither endpoint
Notation example[a, b) or (a, b]Closed: [a, b]; Open: (a, b)
Inequality forma ≤ x < b or a < x ≤ bClosed: a ≤ x ≤ b; Open: a < x < b
Number line graphOne filled circle and one open circleClosed: two filled circles; Open: two open circles
Contains its boundary?Partially — one boundary point onlyClosed: yes, all boundary; Open: no boundary

Why It Matters

Half-open intervals appear frequently in piecewise-defined functions, where each piece covers a range that meets the next piece at a single shared boundary point. They are also the standard way to partition data into bins in statistics and histograms, ensuring every data point falls into exactly one bin. Understanding this notation is essential for reading and writing domain restrictions, solution sets, and function definitions throughout algebra and calculus.

Common Mistakes

Mistake: Mixing up which bracket means included vs excluded — for example, writing (2, 5] when you mean to include 2 and exclude 5.
Correction: Remember: square brackets [ ] mean the endpoint IS part of the interval (think of the bracket "hugging" the number), while parentheses ( ) mean the endpoint is NOT included. To include 2 and exclude 5, write [2, 5).
Mistake: Using a square bracket next to infinity, such as writing [3, ∞].
Correction: Infinity is not a real number, so it can never be reached or included. Always use a parenthesis next to ∞ or −∞. The correct notation is [3, ∞).

Related Terms