Group/Examples/x+y over 1+xy

Theorem
The set of real numbers $$\left\{{x \in \reals: -1 < x < 1}\right\}$$, under the operation:

$$x \circ y = \frac {x + y} {1 + x y}$$

is a group.

Proof
Let $$-1 < x, y, z < 1$$.


 * Check for associativity:

$$ $$

$$ $$ $$


 * Check for identity:

$$ $$ $$ $$

Similarly, putting $$y = 0$$ we find $$x \circ y = x$$.

So $$0$$ is the identity.


 * Check for inverses: put $$y = -x$$:

$$ $$

Similarly, putting $$x = -y$$ gives us $$\left({-y}\right) \circ y = 0$$.

So each $$x$$ has an inverse $$-x$$.


 * Check for closure:

First note that: $$-1 < x, y < 1 \Longrightarrow x y > -1 \Longrightarrow 1 + x y > 0$$.

Next:

$$ $$ $$ $$ $$ $$ $$

Finally:

$$ $$ $$ $$ $$ $$ $$

Thus $$-1 < x, y < 1 \Longrightarrow -1 < x \circ y < 1$$, and we see that in this range, $$\circ$$ is closed. Thus the given set and operation form a group.