Is this a correct assertion regarding universal quantification?

Question

Given an arbitrary number of universal quantifiers n and an arbitrary number of domain elements d:

$ \forall x_1, \forall x_2, \space ... , \forall x_n P(x_1, x_2, ..., x_n) $

• Assertion: There are $d^n$ possible combinations for which the predicate P must be true, in order for the formula to be true.

Example:

$ \forall x \forall y P(x, y) $

Domain: {a, b}

n = 2, d = 2

$ 2^2 = 4 $

(a, a), (a, b), (b, a), (b, b)

Answer 1

Yes. This is essentially asking for the $n$-fold cartesian product on $D$:
$D^n = \underbrace{D \times \ldots \times D}_{n \textrm{ times}} = \{\langle d_1, \ldots, d_n \rangle: d_i \in D\}$.

The number of possible such combinations is the cardinality of this cartesin product, which is given by $|D^n| = |D|^{|n|}$.