I'm preparing a task on the Cartesian product of two sets and I have run into the following confusion:
I understand that the Cartesian product is not a commutative operation. Generally speaking, AxB does not equal BxA unless A=B or A or B is the empty set. This is usually easy to explain to students because in the definition of a cartesian product, we define it as an ordered pair, meaning order would matter. However, once we move on from this idea to explain what the product set represents, things get a bit fuzzy. For example, If we think of the 52 cards in a standard deck as a product set, we can define set A as the ranks and set B as the suits. How can we explain to students that while AxB and BxA both represent the 52 cards in a standard deck, the sets AxB and BxA are distinct and disjoint sets?