Let X be a nonempty set and let G be a group. Suppose that f: XG is a function and let g: W(X) → G be the function defined as follows: For every w = xr² € W (X) where x, X and e, € {1,-1} for all j, define g(u) = f(x₁) f(x₂) 1. Show that g(uv) = g(u)g(v) for all u, v € W (X) 2. If u, v € W (X) such that u→v, show that g(u) = g(v). 3. If u, v € W (X) such that uv, show that g(u) = g(v). 4. If 1 is the empty word on X, show that g(1) = 1c where la is the identity of G.