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Field axiom for proof

  1. Aug 28, 2012 #1
    1. The problem statement, all variables and given/known data

    Using only the fi eld axioms, prove that if x,y ε R and x = y then wx = wy.

    2. Relevant equations

    http://mathworld.wolfram.com/FieldAxioms.html

    3. The attempt at a solution

    The solution to this can be solved within 2 lines or so using the field axiom inverses/multiplications ,however, this is one of a couple of assignment questions and it seems a little too easy? Am I thinking about this question correctly?
     
  2. jcsd
  3. Aug 28, 2012 #2
    It's a very easy problem - there is no "trick" to it. I just did this quickly to check it; I got it using distributivity and additive inverses.
     
  4. Aug 28, 2012 #3

    Zondrina

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    Homework Helper

    Notice :

    x = y ⇔ x - y = 0

    So similarly :

    wx = wy
    wx - wy = 0
    w(x-y) = 0
    w0 = 0

    So either w=0 or 0=0. Im pretty sure your question should require that w≠0 and then the rest should be obvious.
     
  5. Aug 28, 2012 #4
    Why should you require w≠0? It's still a true statement if you take any w real.
     
  6. Aug 28, 2012 #5

    Mark44

    Staff: Mentor

    NO!
    The equation above is what the OP needs to show. You can't start off by assuming what you're trying to prove.
     
  7. Aug 28, 2012 #6

    micromass

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    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    I don't think it can be proven from the field axioms. Rather, you need axioms for the equality operator. This is usually handled in logic courses.

    For now, I think it is enough to say that: x and y are the same number, so wx and wy must be the same number as well.

    For a more rigorous approach, you need mathematical logic.
     
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