Answer:
Explanation:
Answer: Fortunately, you don’t have to prove that x or y must be 1 or -1 in order for x*y to not be an element of G.
You only need to prove that x*y is an element of G for any x,y in G. (That’s what it means for * to be an operation on G.)
I haven’t thought yet about how to prove that, but let me give you some thoughts off the top of my head.
We know -1 < x, y < 1. We must show -1 < (x + y) / (xy + 1) and (x + y) / (xy + 1) < 1.
First, let’s show that -1 < (x + y) / (xy + 1).
I want to multiply xy + 1 to both sides, but we need to know whether it’s positive or not, so we know whether to reverse the inequality.
Well, -1 < x, y < 1 implies that xy > -1, which implies that xy +1 > 0.
So we must show that -(xy + 1) < x + y.
That is, let’s show that 0 < x + y + xy + 1. (I don’t know if this is going anywhere useful; I’m just playing around with algebra.)