  Generalizing the notion of a tight almost disjoint family, we introduce the
notions of a {\em tight eventually different} family of functions in Baire
space and a {\em tight eventually different set of permutations} of $\omega$.
Such sets strengthen maximality, exist under $\mathsf{MA} (\sigma {\rm
-linked})$ and come with a properness preservation theorem. The notion of
tightness also generalizes earlier work on the forcing indestructibility of
maximality of families of functions. As a result we compute the cardinals
$\mathfrak{a}_e$ and $\mathfrak{a}_p$ in many known models by giving explicit
witnesses and therefore obtain the consistency of several constellations of
cardinal characteristics of the continuum including $\mathfrak{a}_e =
\mathfrak{a}_p = \mathfrak{d} < \mathfrak{a}_T$, $\mathfrak{a}_e =
\mathfrak{a}_p < \mathfrak{d} = \mathfrak{a}_T$, $\mathfrak{a}_e =
\mathfrak{a}_p =\mathfrak{i} < \mathfrak{u}$ and $\mathfrak{a}_e=\mathfrak{a}_p
= \mathfrak{a} < non(\mathcal N) = cof(\mathcal N)$. We also show that there
are $\Pi^1_1$ tight eventually different families and tight eventually
different sets of permutations in $L$ thus obtaining the above inequalities
alongside $\Pi^1_1$ witnesses for $\mathfrak{a}_e = \mathfrak{a}_p = \aleph_1$.
