Q.If $A = (2, 4)$ and $B = [3, 5)$, then $A \cap B$ is

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: …