I'm preparing a lesson 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?

  • $\begingroup$ Rather like the point $(1,2)$ not being the same as the point $(2,1)$ when drawn on Cartesian axes $\endgroup$ – Henry Nov 18 '14 at 21:17

You might consider whether the 5 of hearts is different from the heart 5 ? There is clearly a bijection between Rank x Suit and Suit x Rank. So A x B is in that sense quite similar to B x A.

What differentiates A x B from B x A is better revealed when A and B have the same elements (rather than for example suits and ranks), but meaning is attached to order of the pairing of the elements of A and B. What about $R^2 = R \times R$, identifiable with the x-y plane ? Now, we can see that $(x, y) \ne (y, x)$ unless $x = y$.

In the most common (ZF) treatment of ordered pairs, for $a \in A $ and $b \in B$ the ordered pair $(a, b) $ is defined as the set {a, {a, b}}. This distinguishes it from the set {b, {a, b}} unless a = b.

In More Detail

In the card example we create sets representing suits and ranks which are disjoint (in English), so that Suits = {C, D, H, S} and Ranks = {2, 3, 4, ,5, 6, 7, 8, 9, 10, J, Q, K, A}. There is a set representing cards, perhaps identified as Cards = {2C, 3C, ...AS} (at my bridge club they have bar codes for the dealing machine). We see there is a bijection between Suits x Ranks and Cards so that a card is identifiable by its suit and rank, and equally there is a bijection between Ranks x Suits and Cards and a card is identifiable by its rank and suit. Mathematically (in ZF set theory), the elements of the Cartesian products Suits x Ranks differ from the elements of Ranks x Suits: they both consist of elements which are sets, but the one looks like {C, {C, 2}} while the other looks like {2, {2, C}}, or in the more conventional ordered pair notation (C, 2) and (2, C). The elements which comprise the two Cartesian products are different so that in set terms, Suits x Ranks $\ne$ Ranks x Suits. They are interpreted to mean the same because:

  1. When we see an ordered pair the fact that Suits and Ranks are disjoint lets us recognize which element of the pair is a suit and which is a rank.
  2. There are bijections from both of the Cartesian products to Cards and we interpret the ordered pair as a card.

But, in maths a lot of ordered pairs we meet are numbers. In the X-Y plane X = {x|x $\in$ R} and Y = {y|y $\in$ R}, so in fact X = Y = R and it follows that X x Y = Y x X = R$^2$ (any pair of numbers e.g. (5, 3.72) exists in R$^2$, is a point in the X-Y plane and a point in the Y-X plane: the three Cartesian products contain the same elements and are therefore the same set). Now to understand the ordered pair we cannot identify which element belongs to which set by its value, we need to know what the order represents, i.e. the first element is a X-value, the second a Y-value.

This leads to a somewhat ironic conclusion that

  1. if A and B are different sets then A x B $\ne$ B x A (in set terms), but we can probably interpret the meaning of (a, b) and (b, a) to be the same thing
  2. if A and B are the same set then in fact A x B = B x A (in set terms), but (a, b) doesn't mean the same as (b, a) unless a = b.
  • $\begingroup$ @Henry only just read your comment: I agree. $\endgroup$ – Tom Collinge Nov 18 '14 at 22:01

Tose are two distinct models for the sets of cards in a deck. It's true that that card sets are the same, but that doesn't mean the models are identical.

In modeling the 5 students in your classroom, you might choose labels $\{1, 2, 3, 4, 5\}$ or labels $\{A, B, C, D, E\}$. That doesn't make the letter $A$ and the number "1" be the same, right? They may correspond to the same student, but they're different entities.

Mathematicians are great at "punning" by identifying isomorphic things, but they're also good at maintaining a distinction between isomorphic things -- and this is one of the latter cases.


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