I'm having trouble understanding this excerpt from Wikipedia, which defines an operation:
Mainly, I don't understand what is meant by $V \subset X_1 \times...\times X_k$. Why does an operation have multiple domains? I thought there was just one domain and one codomain.
Can you please explain the whole passage in easier terms, the notation is kind of confusing and I'm new to mathematical formalism. Thank you.
On
$k$ given in the article varies. For example ordinary addition of natural numbers $+$ is an operation that is binary and for that $k=2$ because you take $n,m$ and then apply $+$ to get $n+m$...similarly taking modulus is an unary operation as for that $k=1$ as you take $x$ and have $|x|$...thus your $k$ depends on the operation you are defining.
Consider the binary operation $F(x,n)=x^n$, where $x\in\Bbb R$ and $n\in\Bbb N$. Then $F$ has the domain $\Bbb{R\times N}$ and codomain $\Bbb R$.
The reason to allow multiple domains is just to make it easier than to require, each time, that the domain is the union of $X_1\cup\ldots\cup X_k$. I don't know where the text is from, or where it is going to [mathematically], but I can see why this might be an easier set up than requiring one domain and keeping track of it all the time.