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Functions and countable sets

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2.14 (I) Prove that an infinite set X is countable if and only if there is a sequence
of all the elements of X which has no repetitions.

(II) Prove that every subset S of a countable set X is itself countable.
(III) Prove that if there is a sequence of all the elements of a set X, possibly
with repetitions, then X is countable.
(IV) If X is countable and is a surjection, prove that Y is countable.

2.15 Prove that if are countable sets, then is also countable.

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This solution is comprised of a detailed explanation to prove that an infinite set X is countable if and only if there is a sequence
of all the elements of X which has no repetitions.

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We know, a set is said to be countable is there exists an injection . Another equivalent definition is, there exists a surjection , then is a countable set. Here is the set of natural numbers.
1. Proof:
(1) If is countable, then there exists an injection . So is a subset of . We can assume . ...

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