Group/Examples/x+y over 1+xy/Isomorphic to Real Numbers/Proof 1

Theorem
Let $G := \set {x \in \R: -1 < x < 1}$ be the set of all real numbers whose absolute value is less than $1$.

Let $\circ: G \times G \to G$ be the binary operation defined as:
 * $\forall x, y \in G: x \circ y = \dfrac {x + y} {1 + x y}$

Proof
To prove $G$ is isomorphic to $\struct {\R, +}$, we need to find a bijective homorphism $\phi: \openint {-1} 1 \to \R$:


 * $\forall x, y \in G: \map \phi {x \circ y} = \map \phi x + \map \phi y$

From Group Examples: $\dfrac {x + y} {1 + x y}$:
 * the identity element of $G$ is $0$
 * the inverse of $x$ in $G$ is $-x$.

This also holds for $\struct {\R, +}$.

This hints at the structure of a possible isomorphism, namely:
 * $(1): \quad$ that it is an odd function
 * $(2): \quad$ that it passes through $0$
 * $(3): \quad$ that it is defined on the open interval $\openint {-1} 1$.

One such function is the inverse hyperbolic arctangent function $\tanh^{-1}$, which is indeed defined on $\openint {-1} 1$ and has the above properties.

Hence:

This demonstrates the homorphism between $\struct {G, \circ}$ and $\struct {\R, +}$.

We have that Real Inverse Hyperbolic Tangent Function is Strictly Increasing over $\openint {-1} 1$.

Hence from Strictly Monotone Real Function is Bijective, $\tanh^{-1}: \openint {-1} 1 \to \R$ is a bijection.

Hence the result that $\struct {G, \circ} \cong \struct {\R, +}$.

Also see

 * Hyperbolic Tangent of Sum