Thread: More proofs of more theorems

Prove each of the following theorems using any addition and multiplication axioms you know.

1) If each of x and y is a number, and y does not equal 0, then x*(1/y) = (x/y).

2) If each of x and y is a number, and y does not equal 0, then -y does not equal 0, (1/-y) = -(1/y), and -(x/y) = ((-x)/y) = (x/(-y)).

Originally Posted by noles2188

Prove each of the following theorems using any addition and multiplication axioms you know.

1) If each of x and y is a number, and y does not equal 0, then x*(1/y) = (x/y).

2) If each of x and y is a number, and y does not equal 0, then -y does not equal 0, (1/-y) = -(1/y), and -(x/y) = ((-x)/y) = (x/(-y)).

Obviously (1) is not a theorem but a definition of x/y.

Now for part (2):

y=/=0 ====> 1/y =/=0 otherwise y*(1/y)=0,but y*(1/y)=1 and hence a contradiction,thus 1/y=/=0.

y=/=0 ======> -y =/=0.otherwise -y*(1/y)=0 but -y*(1/y)= (-1)*y*(1/y)=(-1)*1=-1 and hence a contradiction,thus -y=/=0.

AND the same way we prove 1/(-y)=/=0
Also -y=/=0 ====> (-y)*(1/(-y))=1 = (-1)*(-1)*1=(-1)*(-1)*y*(1/y)=(-y)*(-(1/y)).

hence (-y)*(1/(-y)) = (-y)* (-(1/y)) and multiplying bothy sides by 1/(-y) we get :

........1/(-y) = -(1/y) since (-y)*(1/(-y) =1.


Now : -(x/y)= -[ x*(1/y)]=(-x)*(1/y) = (-x)/y.

Also : -(x/y) = -[x*(1/y)] = x*(-(1/y)) = x*(1/(-y)) = x/(-y)