Class 11 | CBSE | MATHEMATICS
MATHEMATICS | INTERVALS OF SETS

Big Brain Idea (Anchor Thought)

An interval is a group of real numbers lying between two numbers on the number line.

Imagine walking on a straight road from a to b. The question is: Do we include the endpoints or not?

Open Interval

Let a, b ∈ ℝ and a < b.

The set:

\[ \{ y : a < y < b \} \]

is called an open interval and is denoted by:

\[ (a, b) \]

Brain Rule:
All numbers between a and b are included, but a and b are not included.

Closed Interval

The interval which includes both endpoints is called a closed interval.

\[ [a, b] = \{ x : a \le x \le b \} \]

Memory Hook:
Square brackets [ ] mean included.

Half-Open / Half-Closed Intervals

Sometimes only one endpoint is included.

[a, b) = { x : a ≤ x < b }
👉 includes a but excludes b

(a, b] = { x : a < x ≤ b }
👉 excludes a but includes b

Brain Trick:
Round bracket ( ) → not included
Square bracket [ ] → included

Number Line Representation of Intervals 🧠

Intervals as Subsets of ℝ

Intervals give an alternative way of describing subsets of real numbers.

Example:

A = (−3, 5)
B = [−7, 9]

Since every element of A lies inside B:

\[ A \subset B \]

Important Special Intervals

[0, ∞) → set of all non-negative real numbers

(−∞, 0) → set of all negative real numbers

(−∞, ∞) → represents the entire set of real numbers ℝ

Visualise a number line extending endlessly in both directions.

Infinite Nature of Intervals

Every interval contains infinitely many real numbers.

Even the smallest interval has endlessly many points between its endpoints.

Set-Builder Form ↔ Interval Form

Set-builder form:

\[ \{ x : -5 < x \le 7 \} \]

Interval form:

\[ (-5, 7] \]

Similarly,

\[ [-3, 5) = \{ x : -3 \le x < 5 \} \]

The brain understands better when you can switch between forms.

Length of an Interval

The number (b − a) is called the length of the interval.

This applies to:

(a, b), [a, b], [a, b), (a, b]

Important Insight:
Length depends only on endpoints, not on whether they are included.

Final Brain Snapshot

• Intervals describe continuous parts of the real line
• ( ) means excluded, [ ] means included
• Every interval contains infinitely many points
• Set-builder and interval forms are interchangeable
• Length of interval = b − a