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Math Help - An operation * has 2 possible identity element:which one's the real identity element?

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    Senior Member x3bnm's Avatar
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    An operation * has 2 possible identity element:which one's the real identity element?

    The rule to find identity element is laid out on Pinter's Abstract Algebra book like this on page 24:


    "First solve the equation x * e = x for e; if the equation cannot be solved there is no identity element. If it can be solved, it is
    still necessary to check that e * x = x * e = x for any x \in \mathbb{R}. If it checks, then e is an identity element."


    Suppose there is an operation * on \mathbb{R} such that x * y = \left | x + y \right| and we need to find the identity element.

    So according to the above rule and by the definition of the operation *:

    x * e = \left|x + e \right| = x

    so x = x + e or x = - (x + e)

    e = 0 or e = 2x

    Now x * e = x * 0 = \left| x + 0 \right| = \left| x \right| = x

    But  x * e = x * 2x = \left| x + 2x \right | = \left| 3x \right| \neq x

    So can I say that e = 0 is an identity element for the operation * because it fulfills the property of identity element?

    Can anyone kindly tell me if I'm right or wrong to say that e = 0 is an identity element for this operation?
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    Senior Member x3bnm's Avatar
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    Re: An operation * has 2 possible identity element:which one's the real identity elem

    Sorry I got the solution.

    You see, if x = -1 and e = 0 then x * e = \left | -1 + 0 \right | = 1  \neq -1

    So no  e = 0 is not an identity element for the operation * on the set \mathbb{R} in this case.
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  3. #3
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    Re: An operation * has 2 possible identity element:which one's the real identity elem

    Neither can "2x". An "identity element" for an algebraic stucture, set X with operation *, is member of X such that e+ x= x for any element of X. That is, an identity is a single member of x. There is not a different identity element for each element of X. That operation does not have two identities- it has none.
    Last edited by HallsofIvy; August 24th 2012 at 06:03 PM.
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    Senior Member x3bnm's Avatar
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    Re: An operation * has 2 possible identity element:which one's the real identity elem

    Thanks HallsofIvy for clarifying the rule for being a identity element.
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