Mathematics Homework Solutions
Problem
#86563

Functions and countable sets

(See attached file for full problem description with all symbols)

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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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2.14-2.15.doc




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.

Solution Summary

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.

Solution
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