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1.4 Intervals

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Part of NCERT 11 Mathematics

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concept

Intervals Introduction

Sets on the number line

Intervals: Special Subsets of Real Numbers

An interval is a set of real numbers between two endpoints.

Think: A section of the number line.

Intervals are subsets of R\mathbb{R} (real numbers).

Why they matter: Used extensively in calculus, inequalities, and real-world measurements.

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Open & Closed Intervals

Understanding boundaries

Open Interval (a,b)(a, b)

Definition: (a,b)={x:a<x<b}(a, b) = \{x : a < x < b\}

Key: Endpoints aa and bb are NOT included.

Use round brackets ()()

Think: Open doors - you can't touch the doors.

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Half-Open & Infinite Intervals

More interval types

Half-Open Intervals

Type 1: [a,b)[a, b)

Left endpoint included, right NOT included.

[a,b)={x:a≤x<b}[a, b) = \{x : a \leq x < b\}

Type 2: (a,b](a, b]

Left endpoint NOT included, right included.

(a,b]={x:a<x≤b}(a, b] = \{x : a < x \leq b\}

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Interval Length

Measuring intervals

Length of an Interval

For any finite interval with endpoints aa and bb where a<ba < b:

Length = b−ab - a

Doesn't matter if endpoints are included or not!

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practice

Practice: Intervals

Master interval notation

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Is 5 in the interval (5,10](5, 10]?