What is a Half-Open Interval? | Simple Explanation for Class 9

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What is a Half-Open Interval on a Graph?

Grade Level:

Class 9

AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering

Definition

What is it?

A half-open interval on a graph represents a range of numbers where one endpoint is included, and the other endpoint is excluded. It means values can be equal to one boundary but must be strictly less than or greater than the other boundary.

Simple Example

Quick Example

Imagine a cricket match where a batsman scores between 50 and 100 runs. If the score can be exactly 50 but must be less than 100, we write it as [50, 100). The square bracket '[' means 50 is included, and the round bracket ')' means 100 is not included.

Worked Example

Step-by-Step

Let's represent the interval of mobile data usage 'x' that is greater than 1 GB but less than or equal to 3 GB.

Step 1: Identify the lower bound and upper bound. Lower bound is 1 GB, upper bound is 3 GB.

Step 2: Determine which endpoint is included and which is excluded. 'Greater than 1 GB' means 1 GB is excluded. 'Less than or equal to 3 GB' means 3 GB is included.

Step 3: Choose the correct bracket for each endpoint. For 1 GB (excluded), use a round bracket '('. For 3 GB (included), use a square bracket ']'.

Step 4: Write the interval notation. (1, 3].

Step 5: On a number line, draw an open circle at 1 (to show it's excluded) and a closed circle at 3 (to show it's included).

Step 6: Draw a line connecting these two circles to show all values between them are part of the interval.

Answer: The half-open interval is (1, 3].

Why It Matters

Understanding half-open intervals is crucial in fields like Data Science and Computer Science for defining data ranges, memory allocation, or sensor readings. In Economics, they help define price ranges or production limits. Engineers use them to specify tolerances for parts, ensuring machines work correctly.

Common Mistakes

MISTAKE: Using a square bracket when the endpoint should be excluded. For example, writing [5, 10) for 'greater than 5'. | CORRECTION: Always use a round bracket '(' or ')' when an endpoint is strictly excluded (not equal to).

MISTAKE: Using an open circle on a number line for an included endpoint. For example, drawing an open circle at 7 for the interval [2, 7]. | CORRECTION: A closed (filled) circle on a number line represents an included endpoint (like '[' or ']'). An open (unfilled) circle represents an excluded endpoint (like '(' or ')').

MISTAKE: Confusing the order of numbers in the interval. For example, writing (10, 2) instead of (2, 10). | CORRECTION: Always write the smaller number first and the larger number second in interval notation.

Practice Questions

Try It Yourself

QUESTION: Represent the interval 'x is greater than or equal to 10 but less than 20' using interval notation. | ANSWER: [10, 20)

QUESTION: On a number line, you see an open circle at -3 and a closed circle at 5, with a line connecting them. Write this as a half-open interval. | ANSWER: (-3, 5]

QUESTION: A delivery service guarantees delivery within 30 minutes, but not less than 10 minutes from the order time. If 't' is the delivery time in minutes, write this range as a half-open interval and explain what the brackets mean. | ANSWER: (10, 30]. The round bracket '(' at 10 means delivery time 't' must be strictly greater than 10 minutes (10 minutes itself is excluded). The square bracket ']' at 30 means delivery time 't' can be 30 minutes or less (30 minutes is included).

MCQ

Quick Quiz

Which of the following represents 'all numbers greater than 7 and less than or equal to 15'?

[7, 15)

(7, 15]

[7, 15]

(7, 15)

The Correct Answer Is:

Option B is correct because 'greater than 7' means 7 is excluded (round bracket), and 'less than or equal to 15' means 15 is included (square bracket).

Real World Connection

In the Real World

Many mobile app games, like a racing game, might have levels that unlock based on scores. For instance, 'Level 2' might unlock if your score 'S' is (1000, 2500], meaning you need more than 1000 points but can have up to 2500 points to qualify for that level. This uses a half-open interval to define the score range.

Key Vocabulary

Key Terms

INTERVAL: A set of real numbers between two specified endpoints | ENDPOINT: The numbers that define the start and end of an interval | INCLUDED: The endpoint value is part of the set, represented by a square bracket '[' or ']' or a closed circle | EXCLUDED: The endpoint value is not part of the set, represented by a round bracket '(' or ')' or an open circle | NUMBER LINE: A visual representation of real numbers as points on a line

What's Next

What to Learn Next

Now that you understand half-open intervals, you're ready to explore open intervals and closed intervals. These concepts build on the idea of included and excluded endpoints, helping you represent all types of number ranges accurately on a graph.