# Bounding and dominating numbers for preordering $\leq^*$ on bijections $f:\omega\to\omega$

For functions $$f, g:\omega\to\omega$$ we write $$f \leq^* g$$ if $$\{x\in\omega: f(x)> g(x)\}$$ is finite.

Let $$S_\omega$$ denote the collection of bijections $$\varphi:\omega\to\omega$$ Similarly to the bounding number and the dominating number respectively, we define

$${\frak b}^{\text{bij}} = \min\{|B|: B\subseteq S_\omega \land \forall f\in S_\omega\; \exists b\in B(b\not\leq^* f)\}$$, and

$${\frak d}^{\text{bij}} = \min\{|D|: D\subseteq S_\omega \land \forall f\in S_\omega\; \exists d\in D(f\leq^* d)\}$$.

Do we have $${\frak b}^{\text{bij}}={\frak b}$$? And what about $${\frak d}^{\text{bij}}={\frak d}$$?

Of course $$\mathfrak{b}^\mathrm{bij} \geq 2$$ and it is not hard to see that $$\{id_\omega,f\}$$ is an unbounded family if $$f$$ is the function that permutes each even number with its successor. So $$\mathfrak{b}^\mathrm{bij} =2$$.
I suspect $$\mathfrak{d}^\mathrm{bij}= \mathfrak{c}$$ but I don´t have time to check this.