I love the idea of presenting the different types of multiplication together.
However, the "multiplying two sets" section is not quite right. Firstly, I've never heard it called the "cross product" of two sets, mostly it's just the product and occasionally the "direct product".
More seriously, though, is that it is defined as the set of *ordered* pairs. Using braces {} implies unordered pairs and the language used also implies unordered pairs: "each possible combination of members, one from each".
This becomes particularly important when the two sets have non-empty intersection. With unordered pairs, if a,b are in the intersection then you can't distinguish between {a,b} and {b,a} but they are different elements in A x B. There's also issues with {a,a}.
It does mean that A x B and B x A are not *equal*, but they are naturally isomorphic via the obvious swap. The fact that they aren't actually equal is important in category theory.
Interestingly, the "sum" of two sets is defined as the disjoint union and then distributivity works in that A x (B + C) is naturally isomorphic to A x B + A x C (taking cardinality converts the natural isomorphism to an equality, giving the distributivity law for numbers).

## Dr A Stacey

## 4th Oct 2019 Flag Comment

I love the idea of presenting the different types of multiplication together. However, the "multiplying two sets" section is not quite right. Firstly, I've never heard it called the "cross product" of two sets, mostly it's just the product and occasionally the "direct product". More seriously, though, is that it is defined as the set of *ordered* pairs. Using braces {} implies unordered pairs and the language used also implies unordered pairs: "each possible combination of members, one from each". This becomes particularly important when the two sets have non-empty intersection. With unordered pairs, if a,b are in the intersection then you can't distinguish between {a,b} and {b,a} but they are different elements in A x B. There's also issues with {a,a}. It does mean that A x B and B x A are not *equal*, but they are naturally isomorphic via the obvious swap. The fact that they aren't actually equal is important in category theory. Interestingly, the "sum" of two sets is defined as the disjoint union and then distributivity works in that A x (B + C) is naturally isomorphic to A x B + A x C (taking cardinality converts the natural isomorphism to an equality, giving the distributivity law for numbers).