In Hausdorff's article "Über zwei Sätze von G. Fichtenholz und L. Kantorovich''(1935) one can find the (simplified) proofs of the following two theorems:

1) There are continuum many *essentially different functions* (in the original version they are called *wesentlich verschieden*) from $\omega$ to $\omega$: i.e. there is some $H \subseteq {^\omega \omega}$ such that $|H|=2^{\aleph_0}$ and for every finitely many different $f_0, \dots, f_l \in H$ there is a position $a \in \omega$ such that $f_i(a) \neq f_j(a)$ for $i<j \leq l$.

2) There is an *independent family of subsets* of $\omega$ of size continuum: i.e. there is a family $F \subseteq P_{=\omega}(\omega)$, of infinite subsets of $\omega$, such that $|F|=2^{\aleph_0}$ and for every finitely many pairwise different $x_1, \dots, x_l, y_1, \dots, y_m \in F$ the intersection $x_1 \cap \dots \cap x_l \cap (\omega \setminus y_1) \cap \dots \cap (\omega \setminus y_m)$ is infinite.

I found in Blass and Irwin's paper "Special families of sets and Baer-Specker Groups" that

3) There is an *independent family of functions* on $\omega$ of size continuum: i.e. there is a family $G \subseteq {^\omega}{\omega}$, such that $|G|=2^{\aleph_0}$ and for every pairwise different $f_0, \dots, f_l \in G$ and every (not necessarily distinct) $n_0, \dots, n_l \in \omega$ there is some $a \in \omega$ such that $f_0(a)=n_0, f_1(a)=n_1, \dots, f_l(a)=n_l.$

Blass writes that the proof of 3) is generated by 1) or 2). I cannot see it and I can't find any explicit proof of 3). So I tried (apparently without using directly points 1) and 2)) to find a function $G: 2^{\aleph_0} \times \omega \rightarrow \omega$ such that for every finitely many $(\alpha_0,n_0), \dots, (\alpha_i,n_i) \in 2^{\aleph_0} \times \omega$ with different first coordinates, there is some $k \in \omega$ such that $G(\alpha_j,k)=n_j$ for all $j \leq i$ (the existence of such a $G$ is mentioned by Blass in a discussion in Mathoverflow about iterations of $\sigma$-centered forcings).

Actually I can construct a perfect tree $T\subseteq {^{<\omega}\omega}$ where every natural number appears infinitely many times in each branch and I define $F: [T] \times fin({^{<\omega}}\omega, \omega) \rightarrow \omega$ such that for every $i \in \omega$, for finitely many different branches $f_0, \dots, f_i \in [T] $ and every $n_0, \dots, n_i \in \omega$ there is a $g \in fin({^{<\omega}}\omega, \omega)$ such that $F(f_j, g) = n_j$ for $ j \leq i$. (Elements of $fin({^{<\omega}}\omega, \omega)$ are finite functions from finite sequences of $\omega$ to $\omega$. The function $F$ is defined as $ F(f,g):= f(min\{l: g(f \upharpoonright l) = f(l)\})$ if $ \{l \in \omega: g(f \upharpoonright l)=f(l)\} \neq \emptyset)$ )

I will skip the details because my questions are more related to the following perspective:

a) Why (a modification of) the words *essentially different functions* is not used in 3)?

b) The terminologies *independent family of functions* and *independent family of subsets* are very similar. So can an independent family of subsets generate an independent family of functions of the same size?

c) Is there a reference (book or paper) where one can find an explicit proof of point 3)?

d) Is there a short way to see that 1) or 2) generate 3), as Blass and Irwin claim?