# Peano's Postulates Proofs

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.

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Is this homework? If so, please add the homework tag. –  Ben Crowell Oct 14 '12 at 22:22
@BenCrowell homework tag added, sorry. –  Jakemmarsh Oct 16 '12 at 21:24

Use induction on $a$ for the first. Use case analysis (4 cases: $a = Z$ or $Sx$, $b = Z$ or $Sy$) for the second.