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Math Help - Proof of interesting equation: Related Topics, groups, semigroups, binary operations

  1. #1
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    Question Proof of interesting equation: Related Topics, groups, semigroups, binary operations

    Here is the question:

    "Let G= Z x Q (Where Z is the set of integers and Q is the set of rational numbers) and define a binary operation o by



    (a,b) o (c,d) = (a+c,2-cb+d) - (*)

    Is (G,o)

    (i) a semigroup,
    (ii) a group?

    You must give full reasoning for your answer in order to get full marks."

    [13 Marks]
    ______________________________________
    I require a natural proof to this and I am still a little new to this high level of maths so please excuse me for obvious errors.
    I believe that I understand what the terms in the equation mean e.g. (binary operation - two elements with a calculation applied will produce an element in the same
    set)
    (semigroup - a set with associated binary operation, in this case o, that meets the condition a o (b o c) = (a o b) o c
    (Group - a binary operation attatched to a set)

    I don't understand how we are to apply the above principles to equation (*).

    Any help to shed some light on this would be greatly appreciated.
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  2. #2
    MHF Contributor
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    Re: Proof of interesting equation: Related Topics, groups, semigroups, binary operati

    Hey swcs.

    For a group we have four axioms: 1) Closure 2) Identity 3) Inverse and 4) Associativity. Can you show these?
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