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    Isomorphism of binary structures

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    Determine whether the mapping phi: Z -> Z which is defined by phi(n) = n + 1 is an isomorphism from the binary structure (Z, +) to the binary structure (Z, +). If not, explain why and give a counter-example.

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    Solution Preview

    Note that the two binary structures are identical, since they are both (Z, +).

    Claim: The mapping defined by phi(z) = z + 1 (for z in Z) is not an isomorphism, because phi does not preserve the function "+".

    Proof of Claim: To show that phi does not preserve the function "+", we provide a counter-example, i.e., we produce integers z_1 and z_2 such that ...

    Solution Summary

    A detailed solution of the problem is provided. The given mapping is explored in relation to the given binary structures, and it is determined whether the mapping is an isomorphism. (If it is not an isomorphism, the reason for that is explained and a counter-example is given.)