The set $\{x \in \mathbb{R} : 1 \le x < 2\}$ can be written as ______________.

> [!TLDR] > This is an interval containing $1$ but not $2$; in interval notation it's written as **$\boxed{[1, 2)}$**. Set-builder notation describes a set by stating the properties its elements must satisfy. Interval notation, on the other hand, gives a compact way to write sets of real numbers between two endpoints, using brackets to indicate whether each endpoint is included or excluded. The key is to translate the inequality conditions into the standard interval form: - A square bracket $[$ or $]$ means the endpoint **is included** (corresponding to $\le$ or $\ge$) - A parenthesis $($ or $)$ means the endpoint **is excluded** (corresponding to $ $) Let's decode the given set step by step. 1. **Identify the lower bound**: The condition $1 \le x$ tells us that $x$ can equal $1$ or be larger. Since $1$ is included, we use a square bracket on the left: $[1$. 2. **Identify the upper bound**: The condition $x < 2$ tells us that $x$ must be strictly less than $2$, so $2$ itself is not in the set. Since $2$ is excluded, we use a parenthesis on the right: $2)$. …

The set \x ∈ mathbbR : 1 ≤ x < 2\ can be written a – Class 11 Mathematics

NCERT Exemplar · Q44

\{x \in \mathbb{R} : 1 \le x < 2\}

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Concept understanding — Interval Notation

Interval Notation: A First Look

Imagine you're describing a range of numbers on the number line. Instead of listing every number (which is impossible — there are infinitely many!), we use a shorthand called interval notation.

🧠 The Intuition: "Between" and "Up To"

Think of a stretch of the number line.

Closed interval: "From 2 to 5, including both 2 and 5."

→ You can touch the endpoints.

→ Written as: $[2, 5]$
Open interval: "From 2 to 5, not including 2 or 5."

→ The endpoints are "holes" — you can get infinitely close, but not touch.

→ Written as: $(2, 5)$
Half-open intervals: "From 2 to 5, including 2 but not 5" (or vice versa).

→ Written as: $[2, 5)$ or $(2, 5]$

📐 The Precise Definition

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

We use brackets to show whether the endpoint is included.

Type	Notation	Meaning	Number line picture
Closed	$[a, b]$	$a \le x \le b$	●———●
Open	$(a, b)$	$a < x < b$	○———○
Half-open (left)	$[a, b)$	$a \le x < b$	●———○
Half-open (right)	$(a, b]$	$a < x \le b$	○———●

Key symbols:

[ or ] = included (closed dot on number line)
( or ) = not included (open dot on number line)

♾️ Infinite Intervals

What if the range goes on forever? We use the symbol $\infty$ (infinity).

"All numbers greater than 3" → $(3, \infty)$

(3 is not included, and there's no upper bound)
"All numbers less than or equal to -1" → $(-\infty, -1]$

(no lower bound, -1 is included)

Important exam rule:

Infinity is always paired with a parenthesis ( or ), never a bracket — because infinity is not a number you can "include."

🧪 Quick Examples …

Why this formula?

Interval Notation: Why It Works the Way It Does

Interval notation is a shorthand for describing sets of real numbers that lie between two endpoints. The "why" comes from understanding what the symbols mean in terms of inequalities and the real number line.

1. The Core Idea: Representing a Continuous Range

A real number line is continuous — between any two numbers, there are infinitely many others. Interval notation captures this by specifying:

Where the interval starts (left endpoint)
Where it ends (right endpoint)
Whether the endpoints are included or excluded

The key formulas are just compact translations of inequality statements.

2. The Four Basic Types — Why Each Symbol Is Used

(a) Closed Interval: $[a, b]$

Inequality form: $a \leq x \leq b$

Why the square bracket?

The square bracket [ or ] means "include this endpoint".

$x = a$ is allowed
$x = b$ is allowed
Every number between them is allowed

Derivation:

The set is $\{ x \in \mathbb{R} \mid a \leq x \leq b \}$ .

The square bracket visually "closes off" the endpoint — like a fence that includes the post.

(b) Open Interval: $(a, b)$

Inequality form: $a < x < b$

Why the parenthesis?

The parenthesis ( or ) means "exclude this endpoint".

$x = a$ is not allowed
$x = b$ is not allowed
Only numbers strictly between are allowed

Derivation:

The set is $\{ x \in \mathbb{R} \mid a < x < b \}$ .

The parenthesis is like an open circle on the number line — the endpoint is not part of the set.

(c) Half-Open (or Half-Closed) Intervals: $[a, b)$ and $(a, b]$

Inequality forms:

$[a, b)$ means $a \leq x < b$
$(a, b]$ means $a < x \leq b$

Why mixed symbols?

Each endpoint is treated independently:

Square bracket at the included end
Parenthesis at the excluded end

Derivation:

These arise naturally when one boundary condition is strict and the other is not. For example, "all numbers from 0 up to but not including 5" is $[0, 5)$ .

3. Infinite Intervals — Why the Symbol $\infty$ Gets a Parenthesis

$(a, \infty)$ and $[a, \infty)$

Inequality forms:

$(a, \infty)$ means $x > a$
$[a, \infty)$ means $x \geq a$

Why always a parenthesis at $\infty$ ?

$\infty$ is not a real number — it's a concept meaning "unbounded above".

You cannot "include" infinity because no real number equals infinity
Therefore, the parenthesis is mandatory: $(-\infty, b]$ or $(a, \infty)$

Derivation:

The set $\{ x \in \mathbb{R} \mid x > a \}$ has no largest element. Writing $[a, \infty]$ would falsely suggest $\infty$ is a number that can be reached.

4. Union of Intervals — Why We Use $\cup$

When a set consists of separate pieces, we combine intervals with the union symbol $\cup$ .

Example:

All real numbers except $x = 2$ is written as: …

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This is an interval containing $1$ but not $2$ ; in interval notation it's written as $\boxed{[1, 2)}$ .

Set-builder notation describes a set by stating the properties its elements must satisfy. Interval notation, on the other hand, gives a compact way to write sets of real numbers between two endpoints, using brackets to indicate whether each endpoint is included or excluded.

The key is to translate the inequality conditions into the standard interval form:

A square bracket $[$ or $]$ means the endpoint is included (corresponding to $\le$ or $\ge$ )
A parenthesis $($ or $)$ means the endpoint is excluded (corresponding to $<$ or $>$ )

Let's decode the given set step by step.

Identify the lower bound: The condition $1 \le x$ tells us that $x$ can equal $1$ or be larger. Since $1$ is included, we use a square bracket on the left: $[1$ .
Identify the upper bound: The condition $x < 2$ tells us that $x$ must be strictly less than $2$ , so $2$ itself is not in the set. Since $2$ is excluded, we use a parenthesis on the right: $2)$ . …

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Q.The set $\{x \in \mathbb{R} : 1 \le x < 2\}$ can be written as ______________.

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