- #1
issacnewton
- 1,026
- 36
Hi
I am trying to prove that
[tex]P=\{X\in\mathcal{P}(\mathbb{Z^+})\;|\;X\mbox{ is finite }\} [/tex]
is denumerable. Now here is the strategy I am using. Let
[tex]A_n=\{X\in\mathcal{P}(\mathbb{Z^+})\;|\; |X|=n\;\} [/tex]
So [itex]A_n[/itex] are basically sets of subsets of [itex]\mathbb{Z^+}[/itex] with cardinality
n. So we see that
[tex]P=A_0\cup A_1\cup A_2\cdots[/tex]
I am trying to get help of the following theorem which is proved in the section of the book
(Velleman's How to prove it, section 7.2 )
Theorem: If [itex]\mathcal{F}[/itex] is a family of sets, [itex]\mathcal{F}[/itex] is countable
, and also every element of [itex]\mathcal{F}[/itex] is countable , then [itex]\bigcup\mathcal{F}[/itex] is countable .
So in this direction , let's define a family of sets [itex]\mathcal{F}[/itex]
[tex]\mathcal{F}=\{A_n\;|\;n\in \mathbb{N}\} [/tex]
[tex]\therefore \bigcup\mathcal{F}= P[/tex]
So to use the theorem quoted, I need to prove first that for all n in N, [itex]A_n[/itex] is
countable. How would I go about this ?
Thanks
I am trying to prove that
[tex]P=\{X\in\mathcal{P}(\mathbb{Z^+})\;|\;X\mbox{ is finite }\} [/tex]
is denumerable. Now here is the strategy I am using. Let
[tex]A_n=\{X\in\mathcal{P}(\mathbb{Z^+})\;|\; |X|=n\;\} [/tex]
So [itex]A_n[/itex] are basically sets of subsets of [itex]\mathbb{Z^+}[/itex] with cardinality
n. So we see that
[tex]P=A_0\cup A_1\cup A_2\cdots[/tex]
I am trying to get help of the following theorem which is proved in the section of the book
(Velleman's How to prove it, section 7.2 )
Theorem: If [itex]\mathcal{F}[/itex] is a family of sets, [itex]\mathcal{F}[/itex] is countable
, and also every element of [itex]\mathcal{F}[/itex] is countable , then [itex]\bigcup\mathcal{F}[/itex] is countable .
So in this direction , let's define a family of sets [itex]\mathcal{F}[/itex]
[tex]\mathcal{F}=\{A_n\;|\;n\in \mathbb{N}\} [/tex]
[tex]\therefore \bigcup\mathcal{F}= P[/tex]
So to use the theorem quoted, I need to prove first that for all n in N, [itex]A_n[/itex] is
countable. How would I go about this ?
Thanks