The splitting number is the cardinal defined by:
$$ \mathfrak{s}=\min\{|S|:S\subseteq[\omega]^\omega\wedge (S \ \text{is a splitting family})\} $$
Where being splitting means: $\forall X\in[\omega]^\omega\exists Y\in S(|X\cap Y|=|X\setminus Y|=\omega)$.
I know that $\mathfrak{s}$ is not countable and $\leq2^{\aleph_0}$.
My problem is I need find a splitting family to justify your definition. Can someone help me?