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Math Help - Solvability, cancellation, identity, and inverse properties

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    Solvability, cancellation, identity, and inverse properties

    Prove that if G is a set with an associative operation which has the solvability and cancellation properties, then it has the identity and inverse properties.

    Solvability: for any elements a, b of G, there is an element c in G with a * c = b, and an element d in G with d * a = b
    Cancellation: for any three elements a, b, c of G, if a * b = a * c, then b = c
    Existence of Identity: there is an element e of G so that for all a in G, e * a = a * e = a
    Existence of Inverse: for any a in G, there is an element b in G so that a * b = b * a = e
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Zennie View Post
    Prove that if G is a set with an associative operation which has the solvability and cancellation properties, then it has the identity and inverse properties.

    Solvability: for any elements a, b of G, there is an element c in G with a * c = b, and an element d in G with d * a = b
    Cancellation: for any three elements a, b, c of G, if a * b = a * c, then b = c
    Existence of Identity: there is an element e of G so that for all a in G, e * a = a * e = a
    Existence of Inverse: for any a in G, there is an element b in G so that a * b = b * a = e
    What have you tried? What about solving the equation dc=c?
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