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    Check whether the following sets are groups

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    Determine if the following sets with the given operations are groups. Give a proof or counterexample. Please see attached image for further details.

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    Solution Preview

    To determine whether a set is a group we need to check the following axioms:

    1) Closure under the operation;
    2) Associativity of the operation, that is, for any a,b,c in the set (a.b).c = a.(b.c) (here . is the operation)
    3) Existence of the identity element e, such that for any a a.e = e.a = a
    4) Existence of the inverses, that is for any a there is ...

    Solution Summary

    Three examples of sets each equipped with a binary operation checked for being a group with respect to that operation, by checking the axioms directly.