1. The problem statement, all variables and given/known data See attachment, problem #19a. 2. Relevant equations 3. The attempt at a solution a) Let j ∈ S X S be arbitrary. Then j is an ordered pair of the form (a,b) for some a,b ∈ S. Now let c = a + b + ab ∈ S. Then clearly a*b = c. Now let d ∈ S and assume a*b = d. But then it follows that c = a + b + ab = d. ^My only concern with this proof is that it doesn't explicitly demonstrate that * is closed on S. That is, that a*b is always not -1. I figured this may not have to be shown or just has to do with the properties of addition on the set S, but thought I'd inquire if this is a valid qualm?