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**1. Homework Statement**

Determine (with proofs) which of the following infinite sets are countable and which are uncountable:

(i ) The set of all triples (x, y, z) where x, y, and z are rationals;

(ii ) The set of all subsets of N;

(iii ) The set of all finite subsets of N.

Note: N is Natural Numbers

**2. Homework Equations**

I think there are no relevant equations for this question

**3. The Attempt at a Solution**

For (i), There is a theorem that states all rational sets are countable, so I think it is countable is this right ? If so, I don't know how to write the correct proof.

For (ii), I think it is uncountable becasue the power set of a set S has strictly greater cardinality than S. Is this right, again I don't know how to write the proof for this one.

For (iii), I think it is countable because all sets, constituting of elements from Z (or any countable set), but where an element can occur multiple times (but only finitely many times), is also countable (so these are like subsets, except elements can occur more than once). Is this right, again I don't know how to write the proof for this one.

This is all I can do, can someone help me please ?

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