Zero-ary cartesian product

Question

Let $A=(A_1,...,A_n)$ be a vector of sets. The Cartesian product in $A$ could be defined as:

$\times_A :=\left\{(a_1,..,a_n)\mid \forall a_i \in A_i\right\}$

Which one is correct?

a) $\times_{\emptyset} = \emptyset$ (empty set)

b) $\times_{\emptyset} = \left\{(\ )\right\}$ (set containing the empty vector)

I think b) is correct because the empty $\forall$-operator is always true.

Answer 1

b) is the correct answer. The all-operator evaluates to true of empty.