What is a Half-Open Range?

A half-open range is an interval that includes its starting boundary but excludes its ending boundary. This type of range is widely used in mathematics and computer science to define a sequence or a set of values.

Understanding Half-Open Ranges

The term "half-open" precisely describes its nature: one side of the interval is "closed" (inclusive), while the other side is "open" (exclusive). This convention ensures consistency and simplifies many computational tasks, particularly when dealing with sequences, loops, and array indexing.

Imagine a segment on a number line or a series of items in a list. If you have a range from A to B:

• A is part of the range.
• B is not part of the range.

For example, a half-open range from 0 to 5 would include 0, 1, 2, 3, and 4, but not 5.

Why Use Half-Open Ranges?

The choice of half-open ranges is not arbitrary; it offers several practical advantages, especially in computing:

• Counting Elements: When working with zero-indexed sequences (like arrays or lists in many programming languages), a half-open range [start, end) where start is inclusive and end is exclusive directly provides the count of elements as end - start. For instance, range(0, 5) contains 5 elements (0, 1, 2, 3, 4).
• Looping and Iteration: It simplifies loop conditions. A common pattern is to iterate from i = start up to i < end, naturally fitting the half-open model.
• Concatenation: When combining adjacent half-open ranges, they seamlessly connect without overlaps or gaps. For example, [0, 5) followed by [5, 10) perfectly covers [0, 10).
• Consistency: Many programming language constructs and mathematical definitions adopt this convention, leading to more predictable and less error-prone code.

Half-Open Ranges in Practice

Half-open ranges are prevalent in various programming contexts:

• Python's range() function: Generates a sequence of numbers starting from the first argument up to, but not including, the second argument.
  • list(range(1, 5)) results in [1, 2, 3, 4].
• Array/List Slicing: Many languages use half-open intervals for extracting sub-sections of arrays or strings.
  • In Python, my_list[0:3] includes elements at index 0, 1, and 2, but excludes the element at index 3.
  • In JavaScript, arr.slice(0, 3) similarly creates a new array including elements at index 0, 1, and 2.
• Loop Conditions: for (let i = 0; i < n; i++) is a classic example of using a half-open range for iteration, where i goes from 0 up to n-1.

Notation for Half-Open Ranges

In mathematics, interval notation is used to represent ranges. For a half-open range that includes a but excludes b, the notation is typically [a, b) or [a, b[.

• The square bracket [ indicates inclusion (inclusive boundary).
• The parenthesis ) or [ indicates exclusion (exclusive boundary).

For a more comprehensive understanding of interval notation, you can refer to resources like Wikipedia on Intervals (mathematics). For specific programming examples, consult documentation like the Python range() documentation.