I got a little confused about Cartesian multiplication. Until now, I mainly used Cartesian multiplication over relations, and today my professor started combinatorics. He showed us that performing Cartesian multiplication 'n' times over the same group for example group A results in (a1, a2, a3....,an) where each element is from the group(to prove something in the size of groups). However, when we did relations over NXN, for example, we would get ordered pairs in the structure of ((a1, a2), (a3, a4)).
I would be glad if someone could help me understand the difference between the two and if in the one with relations, the brackets are used only for convenience and does it mean the same as (a1,a2,a3,a4)?