Unbounded Set of Numbers

High School

Definition

Set of numbers that do not have either a lower or upper bound. Bounded sets are finite, whereas unbounded sets can be considered not. To demonstrate this, the sequence of 1, 2, 3, 4, 5... does not have an upper bound to it and is therefore an unbounded set of numbers.

Worked examples

\(\{1, 2, 3, 4, 5, \ldots\}\)

This set grows forever without an upper bound, making it unbounded above.

\(\{\ldots, -3, -2, -1, 0\}\)

This set extends infinitely in the negative direction, so it is unbounded below.

\(\mathbb{Z} = \{\ldots, -2, -1, 0, 1, 2, \ldots\}\)

The integers have no upper or lower bound, making them unbounded in both directions.

Common mistakes

\(\{1, 2, 3, \ldots, 1000\}\) is unbounded because it has many elements → \(\{1, 2, 3, \ldots, 1000\}\) is bounded; it stops at 1000 Unbounded means no limiting value exists, not that the set is large.
All infinite sets are unbounded → \(\{0, \frac{1}{2}, \frac{3}{4}, \frac{7}{8}, \ldots\}\) is infinite but bounded by 0 and 1 A set can be infinite yet stay within finite bounds.
Unbounded sets must extend infinitely in both directions → \(\{1, 2, 3, \ldots\}\) is unbounded even though it has a lower bound of 1 A set is unbounded if it lacks either an upper or a lower bound, not necessarily both.

Where you'll use it next

You'll use unbounded sets when studying sequences and series, analyzing limits and convergence, graphing functions with infinite domains or ranges, and understanding concepts like infinity in calculus.

Found in 1 StudyPug lesson

Set Builder Notation: The Key to Precise Set Descriptions

Geometry

Unlock the power of set builder notation to express complex mathematical sets concisely. Master this essential tool for advanced math and enhance your problem-solving abilities.

10th Grade10th