(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

*Attached is the problem statement, along with a definition which is to be used (I feel like some of the definitions in this text are somewhat untraditional, so I am including this one for clarity).

Edit* sorry, it's problem #31.

2. Relevant equations

3. The attempt at a solution

Let <T, !> be any binary structure isomorphic with <S,*>. This means there is some bijection Ψ: S → T such that (∀s_{1},s_{2}∈ S)(Ψ(s_{1}*s_{2}) = Ψ(s_{1})!Ψ(s_{2})). Now let d ∈ T. Since Ψ is surjective, there is some c ∈ S such that Ψ(c) = d. But then there is some x ∈ S such that x*x = c. As a result, Ψ(x*x) = Ψ(c) = d = Ψ(x)!Ψ(x).

Q.E.D.

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# Homework Help: Anyone up to double check an isomorphism proof?

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