Why is the coproduct denoted by "plus" and the product denoted by "times"?

Question

I found out today on Internet about the notions of products and coproducts. I was wondering why the product was written with the "times" sign and the coproduct was written with the "plus" sign. Please explain simply because I just started reading about this stuff and I basically know nothing yet.

Answer 1

One explanation:

• The product in the category of sets is the cartesian product. And $|A\times B| = |A| \times |B|$.

Indeed, this is why $\times$ is a good notation for the cartestian product of sets!

• The coproduct in the category of sets is the disjoint union. And $|A\sqcup B| = |A| + |B|$.

This suggests that $+$ would be a very good notation for disjoint union of sets. But it's more common to write $\sqcup$ in analogy with the usual union symbol $\cup$. Actually, it's also quite common in category theory to see the notation $\sqcup$ for coproducts in arbitrary categories, borrowed from the notation in the category of sets.