Let $\omega^\omega$ denote the collection of all functions $f:\omega\to\omega$. For $f,g\in\omega$ we define

- $f\leq g$ if $f(n)\leq g(n)$ for all $n\in\omega$;
- $f\leq^* g$ if there is $N\in\omega$ such that $f(n)\leq g(n)$ for all $n\in \omega$ with $n\geq N$.

The cardinals ${\frak b}, {\frak d}$ are defined as follows:

- ${\frak b} = \min\{|S|: S \subseteq \omega^\omega \text{ and } \forall f\in\omega^\omega \exists s\in S(s \not \leq^* f)\}$,
- ${\frak d} = \min\{|S|: S \subseteq \omega^\omega \text{ and } \forall f\in\omega^\omega \exists s\in S(f \leq^* s)\}$.

Let us define ${\frak b}', {\frak d}'$ using $\leq$ instead of $\leq^*$. Can it be proved that ${\frak b}' = {\frak c}$ (which would imply ${\frak d}' = {\frak c}$)? If not: which of ${\frak b}' < {\frak b}, {\frak b}' > {\frak b}, {\frak d}' < {\frak d}, {\frak d}' > {\frak d}$ is consistent?