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Math Help - "Fix" an element, as opposed to "consider" an element?

  1. #1
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    "Fix" an element, as opposed to "consider" an element?

    When doing a proof, is "considering" an arbitrary element equivalent to "fixing" an element? I don't understand the distinction, if any.

    For example, here's a proof from Wikipedia that uses the term "fix."
    If a monoid has the cancellation property and is finite, then it is in fact a group.
    Proof: Fix an element x in the monoid. Since the monoid is finite, x^n = x^m for some m > n > 0. But then, by cancellation we have that x^(m-n) = e where e is the identity. Therefore x * x^(m-n-1) = e, so x has an inverse.
    So in the above proof, if I replace "fix" with "consider," is the proof logically still the same?
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  2. #2
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    Quote Originally Posted by cubrikal View Post
    When doing a proof, is "considering" an arbitrary element equivalent to "fixing" an element? I don't understand the distinction, if any.

    For example, here's a proof from Wikipedia that uses the term "fix."
    So in the above proof, if I replace "fix" with "consider," is the proof logically still the same?

    Yes, it is. In fact, "fix" in this case is probably not convenient since "fixing an element" is generally used when one is going to use a "FIXED" element in several instance and we want to be sure it is always the same FIXED element and not other one.

    Tonio
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  3. #3
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    I agree.

    "fixing an element" is generally used when one is going to use a "FIXED" element in several instance and we want to be sure it is always the same FIXED element and not other one.
    Sometimes people even say, "Consider an arbitrary but fixed element..."
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