Denumerable, in mathematics, describes a set whose elements can be put into a one-to-one correspondence with the natural numbers (1, 2, 3, ...). This means that the set can be enumerated, or listed, in a specific sequence, even if the set contains infinitely many elements. All denumerable sets have the same cardinality (size), known as 'aleph-null' (ℵ₀). Importantly, a denumerable set can be either finite or infinite. This concept is fundamental in set theory and the study of infinity, providing a way to compare the sizes of infinite sets. It helps us understand the relative “size” of infinite sets by classifying them based on their ability to be counted in this specific way. Sets that are not denumerable are considered uncountable.
Denumerable meaning with examples
- The set of even numbers is denumerable. We can list them: 2, 4, 6, 8,... Each even number can be paired with a natural number, showing that the even numbers have the same cardinality as the natural numbers themselves, even though it seems like we are dealing with a smaller section of numbers. This pairing is what makes this set denumerable.
- The set of all integers (positive, negative, and zero) is denumerable. Although seemingly larger than the set of natural numbers, we can still create a one-to-one correspondence: 0, 1, -1, 2, -2, 3, -3,... Thus, even with an apparently greater number of elements, it shares the same size or cardinality as the natural numbers and is still considered denumerable.
- The set of all rational numbers (numbers that can be expressed as a fraction) is denumerable. Even though the rational numbers are dense and appear to fill the number line, we can construct an algorithm, such as a diagonal argument, to list them systematically and create the one-to-one matching with natural numbers. Thus, they're countable.
- A finite set, such as the days of the week, is trivially denumerable because it can be directly paired with natural numbers (1-Monday, 2-Tuesday, etc.). Its denumerability does not suggest anything special as it simply means we can list and order this set of values. This is distinct from infinite sets, which are still considered denumerable.
- Consider an algorithm designed to generate a list of prime numbers. Because each prime number would correspond to a natural number in the algorithm's process, the set of prime numbers is denumerable. The process of generation creates the necessary one-to-one correspondence, showing this set shares cardinality with other denumerable sets.
Denumerable Synonyms
countable countably infinite enumerable listableDenumerable Antonyms
non-denumerable uncountableDenumerable Crossword Answers
9 Letters
COUNTABLE NUMERABLE
10 Letters
ENUMERABLE