Help me find the Cardinality of the following realtions

[pipc]

let \(\displaystyle R\subseteq \mathds{N}^{\mathds{N}}\times\mathds{N}^{\mathds{N}} \)the following relation:
\(\displaystyle R=\{(f,g): each\; i \;satisfies\; f(i)<=g(i)\}\)


for each\(\displaystyle i\in\mathds{N}\) let\(\displaystyle \mathds{N}_{i}\subseteq\mathds{N} \) be the following group:
\(\displaystyle \mathds{N}_{i}=\{0,1,...,i\}\)
and let \(\displaystyle R_{i}=R\cap(\mathds{N}_{i}^{\mathds{N}}x\mathds{N}_{i}^{\mathds{N}})\)

What is the Cardinality of\(\displaystyle R_{0}\) and \(\displaystyle R_{1}\)?

prove\disprove:
a. for each \(\displaystyle i<=j\;\; R_{i}\circR_{j}=R_{j}\)
b. for each \(\displaystyle i<=j \;\;R_{j}\circR_{i}=R_{i}\)

 

[pka]

let \(\displaystyle R\subseteq \mathds{N}^{\mathds{N}}\times\mathds{N}^{\mathds{N}} \)the following relation:
\(\displaystyle R=\{(f,g): each\; i \;satisfies\; f(i)<=g(i)\}\)
for each\(\displaystyle i\in\mathds{N}\) let\(\displaystyle \mathds{N}_{i}\subseteq\mathds{N} \) be the following group:
\(\displaystyle \mathds{N}_{i}=\{0,1,...,i\}\)
and let \(\displaystyle R_{i}=R\cap(\mathds{N}_{i}^{\mathds{N}}x\mathds{N}_{i}^{\mathds{N}})\)

What is the Cardinality of\(\displaystyle R_{0}\) and \(\displaystyle R_{1}\)?

prove\disprove:
a. for each \(\displaystyle i<=j\;\; R_{i}\circR_{j}=R_{j}\)
b. for each \(\displaystyle i<=j \;\;R_{j}\circR_{i}=R_{i}\)

There are some LaTeX errors is the post.
prove\disprove:
a. for each \(\displaystyle i<=j\;\; R_{i}\circ R_{j}=R_{j}\)
b. for each \(\displaystyle i<=j \;\;R_{j}\circ R_{i}=R_{i}\)

Note that \(\displaystyle N_0=\{0\}\) so the cardinality \(\displaystyle \|(N_0)^N\|=1\) WHY?

What does \(\displaystyle \|(N_1)^N\|=~?\)

 

[pipc]

Well I guess that the cardinality of \(\displaystyle \|(N_1)^N\|\) is aleph null

I still don't know what to try and prove with this assignment.
I would appreciate if someone with more knowledge will give me the final answers and I will go and try proving it from there

 

[pipc]

So i'm almost done with this question.

The only thing I still can't figure out is how to prove that R_1 is not countable. I think I have to prove that R_1 cardinality equals P(N) but I am baffled how to formally prove it.

I've tried finding some injective functions and playing with it but didn't really manage.

I'll appreciate some help

 

[pka]

So i'm almost done with this question.
The only thing I still can't figure out is how to prove that R_1 is not countable. I think I have to prove that R_1 cardinality equals P(N) but I am baffled how to formally prove it.

Do you understand that \(\displaystyle 2^{\mathbb{N}}\) is uncountable?
You can use the characteristic function on all subsets.

 

[pipc]

How does the cardinality of 2^N help me prove that the cardinality of R_1 is P(N)? I don't see how they are related.

 

[pka]

How does the cardinality of 2^N help me prove that the cardinality of R_1 is P(N)? I don't see how they are related.

For each \(\displaystyle g\in 2^N\) you know that \(\displaystyle (g,g)\in R_1\).

Thus \(\displaystyle |2^N|\le |R_1|\).

By definition \(\displaystyle R_1\subseteq 2^N\times 2^N\)

Therefore \(\displaystyle |2^N|\le |R_1|\le |2^N\times 2^N|=|2^N|\).

DONE!