I don't know if this is where this type of question should be, so sorry if this isn't it.
I need to show the following using the Field Axioms:
1. That the identity numbers 0 and 1 are unique.
2. That 0*a=0.
3. That the inverses -a and 1/a are unique.
4. That -1*a = -a
5. That (-1)(-1)=1
6. That if , then
Could the first be demonstrated like this?
1. Show that the identity numbers 0 and 1 are unique.
Suppose there is a number , an additive identity, different from zero such that . Then by the definition of additive inverse and by the definition of additive identity which is clearly a contradiction. Therefore, the additive identity is unique.
Suppose there is a number , a multiplicative identity, different from one such that . Then by the definition of multiplicative inverse and by the definition of multiplicative identity which is a contradiction. Therefore, the multiplicative identity is unique.