Prove there is a unique element y such that for all x, x+y=x
I also have to prove their is a unique element y such that for all x, xy=x
The Attempt at a Solution
x+y=x, x+y-x=x-x, y=0. xy=x, xy(1/x)=x(1/x). y=1. The problem with the way I did it is that I had to assume the additive and multiplicative inverses are not defined yet. So I have to prove these statements another way. I am not really sure how to though.
I was given a hint which states that for all x, x+y prime=x. Then consider y+y prime and prove that y=y prime. It didnt really help but when I worked through it I found myself using the additive and multiplicative inverses again which I cant use. How does one go about proving these.