Cardnality of Infinite Sets (4)

[Dick]

After all of these questions, you still can't think of a way to make a correspondence between ternary number is [0,1] and maps from N to {1,2,7} without posting a question and then waiting around for someone to tell you?

 

[kingwinner]

V={{s1,s2,s3,...}|si=0 or 1 or 2}
To each sequence in V, let it correspond to the subset of N consisting of {j|sj=2}
e.g.{1,2,0,1,2,...} <-> {2,5,...}

|V|=|P(N)|=c

But this only shows 2^|N|=c

 

[Dick]

V={{s1,s2,s3,...}|si=0 or 1 or 2}
To each sequence in V, let it correspond to the subset of N consisting of {j|sj=2}
e.g.{1,2,0,1,2,...} <-> {2,5,...}

|V|=|P(N)|=c

But this only shows 2^|N|=c

Sigh. If you meant e.g. {1,2,0,1,2,..}<->{2,7,1,2,7,..}. Yes. No, it DOESN'T show 2^|N|=c. Why would it show that? What does it show?

 

[kingwinner]

Sigh. If you meant e.g. {1,2,0,1,2,..}<->{2,7,1,2,7,..}. Yes. No, it DOESN'T show 2^|N|=c. Why would it show that? What does it show?

OK, I think I get the overall argument:

|{f| f: N -> {1,2,7} }|
=|{sqeuences of 1,2,7}|
=|{sequences of 0,1,2}|
=|{ternary decimals in [0,1]}|
=|[0,1]|
=c

I know this is easy for you, but not so for a student with only a 2-lecture intro to this topic without many practical examples. Thanks for your patience!

 

[Dick]

That's it exactly. You're welcome!