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No subset of the real numbers is countable. True or false.

In looking at the wikipedia article for real numbers, I'm not really clear on the answer to this. They use the words computable and countable, and I'm not sure of the difference. Also, I dont think I fully understand the term countable. The definition states that a set is countable if it has the same cardinality as some subset of the natural numbers, but I'm not sure what that means.

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    Take a finite subset of $\mathbb{R}$ or consider $\mathbb{N} \subset \mathbb{R}$.2010-11-27

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A set is said to be countable if there is bijection between a subset of natural numbers (or even integers) and that set.

http://en.wikipedia.org/wiki/Countable_set

Cantor's diagonalization argument shows that the set of reals is uncountable. So, there are uncountably many subsets of real numbers that are countable. For example the set consisting of any single real number.

Computability has nothing to do with counting the number of elements in a set. This is related to the idea of computable functions.

http://en.wikipedia.org/wiki/Computable_function

A real number is computable if there is finite algorithm that can yield it's digits to any desired level of accuracy.

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    Bijection is "one to one". Injection is "none or one to one". If f (x) = 10*N then f (x) is injective (0,1 to 1) and is definitely not bijective (1 to 1). Surjective is different again. There are three terms because each term has different meaning.2017-07-01
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It is false, because you can always pick a countable subset of your choice (e.g. {1, 2}). A set is countable if it can be put into one-to-one correspondence with the set of Natural numbers or some subset of Natural numbers. Since Natural numbers are contained in Real numbers, you can always pick a subset (like the one mentioned above) that would be countable.