How do I prove the following statement? I have an idea of how to go here, but need a further explanation -
Firstly, I'll define the set of all transitive relation on $\Bbb N$ as $T$.
- First of all, prove that the cardinal number of the set $\mathcal P(\Bbb N\times \Bbb N) = \mathfrak c$. That is easily done.
- Define a function : $f\colon \mathcal P(\Bbb N\times\Bbb N) \to T$, as follows: $R$ is in $\mathcal P(\Bbb N\times \Bbb N)$. If $R$ is transitive, $f(R) = R$, otherwise $f(R) =$ the closest transitive relation that contains $R$.
- That function splits $\mathcal P(\Bbb N\times \Bbb N)$ into different departs, according to the result $f$ sends them to. I need to prove that the cardinal number of those results is $\mathfrak c$.
- There is a $1\leftrightarrow1$ function between $T$ and the group of results of $F$. therefore, cardinal number of $T$ is equal and is $\mathfrak c$.
I can't figure out how to prove point number 3.
(Sorry for my english)