The difference between Countable and Denumerable
When used as adjectives, countable means capable of being counted, whereas denumerable means capable of being assigned a bijection to the natural numbers. applied to sets which are not finite, but have a one-to-one mapping to the natural numbers.
check bellow for the other definitions of Countable and Denumerable
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Countable as an adjective:
Capable of being counted; having a quantity.
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Countable as an adjective (mathematics, of a set):
Finite or countably infinite; having a one-to-one correspondence (bijection) with a subset of the natural numbers.
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Countable as an adjective (mathematics, of a set):
Countably infinite; having a bijection with the natural numbers.
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Countable as an adjective (grammar, of a noun):
Freely usable with the indefinite article and with numbers, and therefore having a plural form.
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Denumerable as an adjective (mathematics):
Capable of being assigned a bijection to the natural numbers. Applied to sets which are not finite, but have a one-to-one mapping to the natural numbers.
Examples:
"The empty set is not denumerable because it is finite; the rational numbers are, surprisingly, denumerable because every possible fraction can be assigned a natural number and vice versa."