1. ## Proofs!

Using any multiplication and addition axioms that you know, prove this theorem.

If each of x and y is a number, then
Part one: (-x)*y = -(x*y) = x*(-y) and
Part two: (-x)*(-y) = x*y.

Part three: Also, if x is a number, then (-1)*x = -x. (-1)*(-1) = 1.

I think it would obviously be easier to do each part separately but I don't know where to start. I'm sure it would be useful to use the definition of minus x in there somewhere, too.

2. Originally Posted by noles2188
Using any multiplication and addition axioms that you know, prove this theorem.

If each of x and y is a number, then
Part one: (-x)*y = -(x*y) = x*(-y) and
Part two: (-x)*(-y) = x*y.

Part three: Also, if x is a number, then (-1)*x = -x. (-1)*(-1) = 1.

I think it would obviously be easier to do each part separately but I don't know where to start. I'm sure it would be useful to use the definition of minus x in there somewhere, too.
$\displaystyle (-x)y + xy = (-x+x)y = 0y = 0 \implies (-x)y = -(xy)$
$\displaystyle x(-y) + xy = x(-y+y) = x0 = 0 \implies x(-y) = -(xy)$

The others follow from these two.