Thread: Need help proving a couple theorems

Prove the following theorems.

1) If x is any number, then 0*x = 0.

2) If x is any number, there do not exist two numbers y and y' such that x*y = 1 and x*y' = 1.

Use any of the following axioms to help prove the above theorems:
1) If each of x and y is a number, then x*y is a number.
2) If each of x and y is a number, then x*y = y*x.
3) If each of x, y, and z is a number, then (x*y)*z = x*(y*z).
4) There is a number U such that if x is any number, then U*x = x.

Originally Posted by noles2188

Prove the following theorems.

1) If x is any number, then 0*x = 0.

this is actually an interesting one. you usually see this as an axiom. not always, but a lot of the time.

....axiom (4)



.....distributive property

.......axiom (4)

2) If x is any number, there do not exist two numbers y and y' such that x*y = 1 and x*y' = 1.

i suppose you mean distinct numbers y and y', of course.

assume to the contrary that there does exist distinct numbers y and y' such that

x*y = 1.........(a) and,
x*y' = 1 ........(b)

subtracting (b) from (a) we get

x*y - x*y' = 0

=> x(y - y') = 0

thus, x = 0 or y = y'

in either case, we have a contradiction. (do you see why? in particular, why does x = 0 yield a contradiction?)