Cardinality of quotient set

Hello,

Here's a relation defined on the set .

is in relation with when

1)What's the cardinality of the quotient set for such that for every

In order for to be in relation with it has to always equal after some time..

1)What's the cardinality of the quotient set for such that for every

For to be in relation with after some time it can't go upwards from some chosen .

I have literally no idea how to even begin assigning cardinal number to those quotient sets. Could someone provide me a hint?

Re: Cardinality of quotient set

Re: Cardinality of quotient set

Thanks, so I no longer have trouble with the second equivalence class, but still have trouble with the first.

I tried proving using Cantor-Bernstein theorem, and I have a function , but have trouble coming up with . Can anyone give me a hint?

Re: Cardinality of quotient set

One of the simplest bijections for k = 2 is . See also Cantor's pairing function.

Re: Cardinality of quotient set

Can it be something like this?

For some finite sequence of length .

EDIT:

It's wrong because ...

Re: Cardinality of quotient set

Re: Cardinality of quotient set

How do we know that a function defined like this is injective?

Re: Cardinality of quotient set

We prove it by induction on k.

Re: Cardinality of quotient set

Say i define this function like this:

For a set of two elements

For elements

So for is obviously injective, because because decomposition into prime factors is unequivocal which means that for each pair a unique would be assigned. So let's assume that is a unique number, not assigned to any other sequence of numbers, therefore is unique for the same reasons the sequence of two elements was unique.

Re: Cardinality of quotient set