Could multiplication be defined as $|A_1 \cup A_2 \cup ..A_n|$?

Question

Taking discrete math which suddenly right now I am thinking of what dividing and multiplication do when it comes to what I learned in this class. But would multiplication be defined as the cardinality of a union of a set and itself? Not sure how to word this properly.

Answer 1

If A is a set such that |A| = m, then you can take the cartesian product of A with itself $n$ times to get $m^n$. $$|A\times A\times...\times A| = |\{(a_1,a_2,...,a_n) : a_j \in A \}| = |A|^n = m^n.$$ If you wanted to get a set of size $mn$, you can take the disjoint union of $A$ with itself $n$ times $$|\cup_{i=1}^n \cup_{j=1}^m \{(i,a_j)\}| = nm $$