I'm curious. I have this set $$\mathbb{R}\setminus\{-1\}$$ and a binary operation defined by: $x*y = x+y + xy$.
How do I prove closure of this operation? It seems obvious to me that that operation yields a real number, but I just don't know how to prove it. I was looking for exactly what a Real number is, so I can use the definition, but I am having trouble finding such a definition.
Suppose $x+y+xy=-1$. This means $$xy+x+y+1=(x+1)(y+1)=0.$$ So either $x=-1$ or $y=-1$.