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Mathematical Foundations for ML

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Mathematical Foundations for ML · 82 lessons

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Interval notation

Bounded intervals

Explanation

An inequality like $2 \leq x < 5$ describes a range of values. Interval notation is a compact way to write these ranges without repeating the variable.

We write the two endpoints separated by a comma, and use brackets to show whether each endpoint is included or excluded.

A square bracket $[$ or $]$ means the endpoint is included (matching $\leq$ or $\geq$ ).
A round bracket $($ or $)$ means the endpoint is excluded (matching $<$ or $>$ ).

This gives us four types of bounded interval:

Interval	Inequality	Name
$[a, b]$	$a \leq x \leq b$	Closed
$(a, b)$	$a < x < b$	Open
$[a, b)$	$a \leq x < b$	Half-open (closed-open)
$(a, b]$	$a < x \leq b$	Half-open (open-closed)

The left endpoint must always be less than the right endpoint. For example, $[3, 7]$ is valid but $[7, 3]$ is not.

Single points and empty intervals

Gotcha

The interval $[a, a]$ contains just one value, $a$ itself, since $a \leq x \leq a$ forces $x = a$ . However, $(a, a)$ is empty because no number is strictly between $a$ and $a$ .

Example

Convert the compound inequality $2 \leq x < 5$ into interval notation. Then convert the interval $(-3, 1]$ back into an inequality.

Solution

Part 1: Inequality to interval

The inequality $2 \leq x < 5$ has two endpoints: $2$ and $5$ .

The left endpoint is $2$ with $\leq$ (included), so we use a square bracket: $[2$ .
The right endpoint is $5$ with $<$ (excluded), so we use a round bracket: $5)$ .

Putting them together:

[2, 5)

Part 2: Interval to inequality

The interval $(-3, 1]$ has endpoints $-3$ and $1$ .

The round bracket at $-3$ means excluded, giving $-3 < x$ .
The square bracket at $1$ means included, giving $x \leq 1$ .

Combining these:

-3 < x \leq 1

Practice questions

4 questions

Which interval notation represents the inequality $-1 < x \leq 4$ ?

Select the correct answer:

$[-1, 4]$

$(-1, 4]$

$[-1, 4)$

$(-1, 4)$

+ 3 more questions

Unbounded intervals

Explanation

Not every range has both endpoints. The inequality $x > 3$ has no upper bound - the values extend indefinitely. We handle this with the symbol $\infty$ (infinity) and its counterpart $-\infty$ .

Key Point

$\infty$ and $-\infty$ are not numbers. They are symbols representing unboundedness. Because we can never reach or equal infinity, it always gets a round bracket - never a square bracket.

The unbounded forms are:

Interval	Inequality	Meaning
$(a, \infty)$	$x > a$	All values greater than $a$ .
$[a, \infty)$	$x \geq a$	All values greater than or equal to $a$ .
$(-\infty, b)$	$x < b$	All values less than $b$ .
$(-\infty, b]$	$x \leq b$	All values less than or equal to $b$ .
$(-\infty, \infty)$	All $x$	Every number on the number line.

The bracket at the finite endpoint follows the same rule as before: square for included ( $\leq$ or $\geq$ ), round for excluded ( $<$ or $>$ ). The bracket next to $\infty$ or $-\infty$ is always round.

Square brackets with infinity is invalid

Gotcha

Writing $[a, \infty]$ with a square bracket at infinity is incorrect notation. Since $\infty$ is not a value that $x$ can equal, it must always be paired with a round bracket.

Example

Express "all real numbers greater than or equal to $-1$ " in interval notation.

Solution

The phrase "greater than or equal to" tells us the inequality is $x \geq -1$ . There is no upper bound - the range extends indefinitely to the right.

The finite endpoint is $-1$ . The inequality uses $\geq$ , so $-1$ is included - we use a square bracket: $[-1$ .
The range has no upper limit, so we write $\infty$ on the right. Since infinity is never reached, it always gets a round bracket: $\infty)$ .

Putting them together:

[-1, \infty)

The square bracket confirms that $-1$ itself belongs to this interval. The round bracket at $\infty$ is mandatory because $\infty$ is not a real number - it cannot be included.

Practice questions

4 questions

Which interval notation represents the inequality $x < 7$ ?

Select the correct answer:

$(-\infty, 7)$

$(-\infty, 7]$

$(7, \infty)$

$[7, \infty)$

+ 3 more questions