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Need some help on a proof!

  1. Dec 12, 2004 #1
    Howdy, I just stumbled on this forum and was hoping someone could help with this proof:

    If a and b are elements of the Rational Number set, then a*b=0, if and only if a=0 or b=0

    With that in mind, I need to prove that:

    • Prove that if a=0 or b=0, then a*b=0
    • Prove that if a*b=0, then a=0 or b=0

    Any help is appreciated, cheers.
  2. jcsd
  3. Dec 12, 2004 #2
    a) One can assume, without loss of generality, that a = 0. Then for any b, we have that ab = 0b = (0 + 0)b = 0b + 0b. Subtract 0b from both sides and you'll find that 0b - 0b = 0b, or equivalently 0b = 0.

    b) Can we assume that every non-zero rational has an inverse? If both a and b are zero, then we are done. Suppose a is non-zero. Then b = 0*a^-1 = 0. The same argument can be repeated if b is non-zero.
    Last edited: Dec 12, 2004
  4. Dec 12, 2004 #3


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    Staff Emeritus
    Science Advisor

    One of the "axioms" or defining properties of the rational numbers is that every rational number, except 0, has a multiplicative inverse.

    Given that ab= 0, either
    1) a= 0 in which case we are done, or

    2) a is not 0, in which case a-1(ab)= a-10 or b= 0.
  5. Dec 12, 2004 #4
    Thanks for the help!
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