Proving Theorems with Axioms

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• Feb 16th 2011, 06:03 PM
>_<SHY_GUY>_<
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), \$\displaystyle (-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.
• Feb 16th 2011, 06:25 PM
Tinyboss
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.
• Feb 16th 2011, 06:30 PM
>_<SHY_GUY>_<
Quote:

Originally Posted by Tinyboss
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.
• Feb 16th 2011, 06:36 PM
Tinyboss
Use the distributive property to continue: 0 = a(1-1) = a(1+(-1)) = a*1 + a*(-1) = a + a*(-1).