What is a Closed Interval?

S3-SA5-0163

Grade Level:

Class 10

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

Definition

What is it?

A closed interval is a set of all real numbers between two specific numbers, including those two numbers themselves. Think of it as a tightly 'closed' section on the number line, where the start and end points are part of the interval.

Simple Example

Quick Example

Imagine a shop selling mangoes where the price per kg is fixed between Rs 80 and Rs 120, including both Rs 80 and Rs 120. This range, [80, 120], represents a closed interval, meaning you can pay Rs 80, Rs 120, or any price in between.

Worked Example

Step-by-Step

PROBLEM: Represent the set of all real numbers x such that x is greater than or equal to 3 and less than or equal to 7, using interval notation.
---STEP 1: Identify the lower bound. The problem states 'x is greater than or equal to 3', so the lower bound is 3.
---STEP 2: Identify the upper bound. The problem states 'x is less than or equal to 7', so the upper bound is 7.
---STEP 3: Determine if the bounds are included. Since 'greater than or equal to' and 'less than or equal to' are used, both 3 and 7 are included in the set.
---STEP 4: Use square brackets for included bounds. Square brackets [ ] are used to show that the numbers at the ends of the interval are included.
---STEP 5: Write the interval notation. Combine the lower bound, upper bound, and square brackets.
ANSWER: The closed interval is [3, 7].

Why It Matters

Closed intervals are super important for setting boundaries and limits in many fields. In AI/ML, they define valid ranges for data, like image pixel values. In Physics, they help describe the range of possible values for measurements. Understanding them is key for careers in data science and engineering.

Common Mistakes

MISTAKE: Using round brackets ( ) instead of square brackets [ ] for closed intervals. | CORRECTION: Always use square brackets [a, b] to show that 'a' and 'b' are included in a closed interval.

MISTAKE: Confusing the order of numbers, e.g., writing [7, 3] instead of [3, 7]. | CORRECTION: The smaller number always comes first in interval notation, representing the lower bound, followed by the larger number as the upper bound.

MISTAKE: Thinking that only whole numbers are included in a closed interval. | CORRECTION: A closed interval includes ALL real numbers between and including the endpoints, not just integers. For example, [1, 2] includes 1.5, 1.732, etc.

Practice Questions

Try It Yourself

QUESTION: Write the interval notation for all real numbers x such that 0 <= x <= 5. | ANSWER: [0, 5]

QUESTION: A mobile phone battery charges from 10% to 100%. If 'P' represents the charge percentage, write this range as a closed interval. | ANSWER: [10, 100]

QUESTION: The temperature in a cold storage unit must be kept between -5 degrees Celsius and 2 degrees Celsius, including these temperatures. Express this temperature range using interval notation. | ANSWER: [-5, 2]

MCQ

Quick Quiz

Which of the following correctly represents a closed interval?

(2, 8)

[2, 8)

(2, 8]

2, 8

The Correct Answer Is:

A closed interval means both endpoints are included, which is represented by square brackets [ ]. Options A, B, and C use round brackets, indicating that at least one endpoint is not included.

Real World Connection

In the Real World

When you book train tickets on IRCTC, the system often has a 'minimum age' and 'maximum age' for certain ticket types, like children's fares. For example, children between ages 5 and 12 (inclusive) might get a discount. This age range, [5, 12], is a perfect example of a closed interval in a real-world Indian scenario.

Key Vocabulary

Key Terms

INTERVAL: A set of real numbers between two given numbers | LOWER BOUND: The smallest number in an interval | UPPER BOUND: The largest number in an interval | REAL NUMBERS: All numbers on the number line, including fractions, decimals, and integers | INCLUSIVE: Including the endpoints

What's Next

What to Learn Next

Great job understanding closed intervals! Next, you should explore 'Open Intervals' and 'Half-Open/Half-Closed Intervals'. These build on what you've learned and show you how to represent ranges where the endpoints might or might not be included.