How can I prove the following two questions:
Prove using Peano's Postulates for the Natural Numbers that if a and b are two natural numbers such that a + b = a, then b must be 0?
Prove using Peano's Postulates for the Natural Numbers that if a and b are natural numbers then: a + b = 0 if and only if a = 0 and b = 0?
I understand the basic postulates, but not sure how to apply them to these specific questions. The questions seem so basic and obvious, but when it comes to applying the postulates I am lost.
Use induction on $a$ for the first. Use case analysis (4 cases: $a = Z$ or $Sx$, $b = Z$ or $Sy$) for the second.