My lecture defined that a boolean function is a function f: ${\{0,1\}}^{V} → \{0,1\}$ with $V$ being the set of variables. Moreover a boolean function f: ${\{0,1\}}^{V} → \{0,1\}$ is finite if set of "relevant" variables is finite. A variable is relevant in f if there are two interpretations $I_1,I_2: V → \{0,1\}$ such that $I_1(x) ≠ I_2(x)$ and $f(I_1) ≠ f(I_2)$.
This definition is new to me, as previous lectures defined finite boolean functions as functions: f: ${\{0,1\}}^{n} → \{0,1\}$. Can someone please explain (proof?) how my the second definition derives from the first definition?
Anything that is a finite boolean function under the second definition is also a finite boolean under the first definition. But the first definition is more general. It includes some functions that are not finite boolean functions under the first definition.
When $V$ is finite then we know $|V|=n$ for some number $n$. Since the $n$ in $\{0,1\}^n$ just means “some set of size $n$”, the formulas $\{0, 1\}^n$ and $\{0, 1\}^V$ mean exactly the same thing when $V$ is finite.
The difference arises when $V$ is an infinite set. The $\{0,1\}^n$ formula doesn't consider this case. Let's say that $V$ is the infinite set $$V =\{v_1, v_2,v_3\ldots, v_{10934}, \ldots \}.$$
Now let's consider the function $f : V\to \{0, 1\}$ which is:
$$ f(v_1, v_2, v_3, \ldots) = \begin{cases} 1,\text{ if both $v_1$ and $v_2$ are $1$} \\ 0,\text{ otherwise} \end{cases} $$
$f$ here is a function of an infinite number of variables, but according to the definition, only $v_1$ and $v_2$ matter to the output of the function. $f$ is really behaving just like the very simple function $$f = v_1\land v_2,$$ except it has an infinite family of inputs that never affect its value. You can imagine $f$ as a circuit with an infinite number of input wires. The first two wires are connected to an and-gate, and all the other wires are connected to nothing.
We may as well ignore the inputs that are not connected (they are “irrelevant”) and just say that $f$ is also a finite boolean function that has two relevant inputs.
Some functions on an infinite set of variables are not finite boolean functions under this definition. For example:
$$ f(v_1, v_2, v_3, \ldots) = \begin{cases} 1,\text{ if exactly three of the $v_i$ are $1$} \\ 0,\text{ otherwise} \end{cases} $$
This function has an infinite number of inputs, and they are all relevant, so it's not a finite boolean function under either definition.