If we consider this equation
$$S+J=11$$
where $S$ and $J$ are Sam and Jane and their combined age is $11$, and I don't allow for one to be $11$ while the other is $0$ years of age, then I have a possible age set for both children of
$$A = \{1,2,3,4,5,6,7,8,9,10\}$$
where the possible solutions is the set of 2-tuples $(a,b)$ (or the set of sets containing combinations of the $A$) where $a+b = 11$. My confusion is that I don't think I can call this a Cartesian product and the possible solutions a set of 2-tuples since both $(a,b)$ and $(b,a)$ work, as long as they add to $11$. I'm confused at how this seems like a Cartesian product but maybe isn't. What could it be if it is not a Cartesian product? Of course the regular set
$$A_{S+J} = \{\{1,10\},\{2,9\},\{3,8\},\{4,7\},\{5,6\}\}$$
works, but then the set
$$A_{S+J} = \{(1,10),(2,9),(3,8),(4,7),(5,6),(6,5),(7,4),(8,3),(9,2),(10,1)\}$$
seems plausible to this beginner as well, since the order does seem to matter, i.e., if Sam is $1$ then Jane must be $10$, and this is not the same as Sam is $10$ and Jane is $1$. But then if I draw a Cartesian coordinate system, I could list the age set $1\ldots10$ along the $x$ and $y$ axes, then draw a negative slope line, albeit once I'm past $5$ on the $x$ axis, I'm essentially repeating myself. I'm confused, although my guess is the latter, i.e., this is indeed a Cartesian product for the reason I stated. But I have nagging doubt because I've got only one set instead of two, although I could just repeat $A$, i.e., create $A_S$ and $A_J$, even though they're the same. I could also use some help expressing this in set notation.
I would go with the latter set $\{(1,10),(2,9),\dots, (10,1)\}$ since it is different for Sam to be 10 and Jane to be 1 than it is for Sam to be 1 and Jane to b 10. This set is not a Cartesian product, i.e., we cannot write it as $X\times Y$ for any sets $X$ and $Y$, but it is a subset of the Cartesian product $A^2$.