Definition

1. Adjective:
   • Capable of being counted; countable: Denumerable describes a set or collection whose elements can be put into a one-to-one correspondence with the natural numbers (1, 2, 3, ...). It implies the set is infinite but countable.
   • Able to be enumerated or listed in a sequence: Denumerable refers to something that can be listed item by item in a systematic order.

Usage Examples

• Adjective:
  • The set of all integers is denumerable, as we can list them as 0, 1, -1, 2, -2, and so on.
  • While the points on a line are not denumerable, the set of all fractions is.
  • The researcher needed to work with a denumerable set of data points for the algorithm to function.

Advanced Usage

• In Mathematics/Set Theory: A set is called denumerable if it has the same cardinality as the set of natural numbers. This is a formal term for a countably infinite set.
  • Cantor proved that the set of rational numbers is denumerable.
• Contrast with "Finite" and "Uncountable": "Denumerable" specifically means . A finite set (like the set of planets) is countable but not denumerable. An uncountable set (like the set of real numbers) is neither.
  • A finite list is countable, but only an infinite list that can be systematically ordered is denumerable.

Variants and Related Words

• Denumerability (n): The property of being denumerable.
  • The denumerability of the algebraic numbers was a key discovery.
• Enumerate (v): To list or count items one by one.
  • Please enumerate your main arguments.
• Numerable (adj): Capable of being counted or numbered (often used for finite collections).
  • The mistakes were few and numerable.

Synonyms

• Countable (adj): Able to be counted.
• Enumerable (adj): Capable of being listed out.
• Countably infinite (adj, Mathematics): Having an infinite number of elements that can be matched with natural numbers.

Antonyms

• Uncountable (adj): Not capable of being counted; infinite in a way that cannot be systematically listed (e.g., the set of real numbers).
• Non-denumerable (adj): Not denumerable; uncountable.

Adjective

1. that can be counted
   • countable sins
   • numerable assets