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**1) Consider the xy-plane.**

Find the cardinality of the set of constructible points on the x-axis.

Find the cardinality of the set of constructible points on the x-axis.

__Attempt:__

Every constructible number is algebraic (i.e. Let A=set of algebraic numbers, C=set of constructible nubmers, then C is a subset of A)

and A is countable.

=> |C|

__<__|A|=|N|

=>|C|

__<__|N|

=> C is either finite OR countably infinite

I belive that C should be an infinite set, but how can I prove this?

Also, is the set {constructible points on the x-axis} an infinite set? How can I prove this? I am stuck here...

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**2) Find the cardinality of the set of all finite subsets of Q.**

__Attempt:__

Let S={all finite subsets of Q}

For every k E N, let A

_{k}= {all subsets of Q having

**EXACTLY**k elements}

S is equal to

∞

U A

_{k}U {empty set}

k=1

k is a natural number, so the union is a union of a countable number of sets.

__Theorem:__The union of a countable number of countable sets is countable.

So by this theorem, S is countable if we can prove that A

_{k}is countable for every k E N.

S is also infinite, so S is countably infinite, i.e. |S|=|N|

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Does this proof work? If so, then it remains to prove that A

_{k}is countable for every k E N, how can we prove this?

Any help would be appreciated! Thank you!