Hello,
Can you help me with proving for integer a,b :
a+b integer
a.b integer
I posted this on number theory but apparently it belongs to set theory
thank you
This is how I would do it:
The successor function is defined as $\displaystyle s: \mathbb{Z}^{+} \rightarrow \mathbb{Z}^{+} $ as $\displaystyle s(n) = n+1 $ for $\displaystyle n \in \mathbb{Z}^{+} $ (for positive integers).
So by the Peano postulates, $\displaystyle a+b $ and $\displaystyle ab $ should be integers.
$\displaystyle a+1 = s(a) $, and $\displaystyle a + s(k) = s(a+k) $.
$\displaystyle a \times 1 = a $ and $\displaystyle a \times s(k) = a \times k +a $.
Correct me if I am wrong.
I think so. I believe you have to use the fundamental theorem of arithmetic for other other method of proof. So write out the unique prime factors of $\displaystyle a,b $ and then add them and multiply them. If you get a unique set of prime factors for each operation then you will have proved that $\displaystyle a+b, ab \in \mathbb{Z} $.