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Math Help - Proving Theorems with Axioms

  1. #1
    Member >_<SHY_GUY>_<'s Avatar
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    Question Proving Theorems with Axioms

    Apologies for the lack of a better way to state the problem I'm having right now.

    I'm trying to prove 2 theorems given by my professor:

    1. ∀a∈ ℝ (Real numbers), (-1)*a = -a

    2. If a, b ∈ (real numbers) and a =/= 0 and b =/= 0, then a*b =/=. If a, b ∈ (real numbers) and a*b = 0 and a=/= 0, then b = 0

    For the first one, I understand that I need to use the axiom which states: 1*a = a, but I don't know how to continue.

    The second one, honestly, confuses me and I don't know where to start.
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  2. #2
    Senior Member Tinyboss's Avatar
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    For the first, use the distributive property: 0=a*0=a(1-1).

    For the second, suppose ab=0, then multiply on the left by 1/a.
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  3. #3
    Member >_<SHY_GUY>_<'s Avatar
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    Quote Originally Posted by Tinyboss View Post
    For the first, use the distributive property: 0=a*0=a(1-1).

    For the second, suppose ab=0, then multiply on the left by 1/a.
    Thank you for the help. I have a question on what you stated.

    How do you approach the step " 0 = a*0 = a(1 -1) ? I'm lost there.
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  4. #4
    Senior Member Tinyboss's Avatar
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    Use the distributive property to continue: 0 = a(1-1) = a(1+(-1)) = a*1 + a*(-1) = a + a*(-1).
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