Now to finish this question, to show a bijection, I will need to show the injection (1-1) and onto-ness (surjection). In order for a function to be 1-1, for all x1,x2 in X x X, f(x1)=f(x2) implies that x1 = x2. And in order for the function to be onto, for all elements, say (x1,x2) in the codomain X x X , there exists (f(1),f(2)) in the domain such that f(f(1),f(2)) = (x1, x2).
Would that be correct?
So, Sigma(alpha) = Sigma (Beta) implies that (alpha(1),alpha(2)) = (beta(1), beta(2)), which implies that alpha = beta.
onto: So, let (a,b) in X x X be arbitrary and let (gamma) = sigma(alpha). Then sigma(gamma)= (gamma(1), gamma(2)) = (a,b)?
Does that follow? Thanks for walking this through, I am definitely learning the process here.