Unbounded Sets

What is an unbounded set?

A set A is considered unbounded if its bounds extend infinitely in both directions: $$ inf(A)=-\infty $$ $$ sup(A)=+\infty $$

• A set is unbounded below if its lower bound is negative infinity (-∞).
• A set is unbounded above if its upper bound is positive infinity (+∞).
• A set is simply called unbounded if both its upper and lower bounds are infinite.

Practical Examples

Example 1

The set of real numbers, R, is unbounded.

$$ inf(R) = -\infty $$

$$ sup(R) = +\infty $$

Since both the lower and upper bounds extend to infinity, R is unbounded.

Example 2

The set of natural numbers, N, is unbounded above.

$$ inf(N) = 0 $$

$$ sup(N) = +\infty $$

In this case, only the upper bound is infinite (+∞), while the lower bound is a finite number (0).

Example 3

The set of negative real numbers, R^-, is unbounded below.

$$ inf(R^-) = -\infty $$

$$ sup(R^-) = 0 $$

Here, only the lower bound extends to infinity (-∞), while the upper bound is a finite number (0).

And so on.