- #1
kingwinner
- 1,270
- 0
1) Consider the xy-plane.
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 believe 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...
=================================
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 Ak = {all subsets of Q having EXACTLY k elements}
S is equal to
∞
U Ak 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 Ak is countable for every k E N.
S is also infinite, so S is countably infinite, i.e. |S|=|N|
==================
Does this proof work? If so, then it remains to prove that Ak is countable for every k E N, how can we prove this?
Any help would be appreciated! Thank you!
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 believe 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...
=================================
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 Ak = {all subsets of Q having EXACTLY k elements}
S is equal to
∞
U Ak 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 Ak is countable for every k E N.
S is also infinite, so S is countably infinite, i.e. |S|=|N|
==================
Does this proof work? If so, then it remains to prove that Ak is countable for every k E N, how can we prove this?
Any help would be appreciated! Thank you!