indirect proof divisor rule?
Good evening,
for the divisor rule
āš,š,š ā Zā¶ š | š and š ā¤ š ā š ā¤(š+š)
Should I use an indirect proof? Unfortunately, I don't know what I should "change" to make it a proof by contradiction.
I would really appreciate any help! Thanks!
Accepted (a | b) and (a | (b+c)), then there is a k,l ā¬ Z with
(I) a*k = b
(II) a*l = b + c
(II) is followed by b = a*k:
a*l = a*k + c
a*l – a*k = c
a*(l – k) = c
This follows (a | c), for (l-k) ā¬ Z
Objection to (a ā¤ c)
From a | b follows the existence of a k ā¬ Z with a * k = b
From a ā¤ c follows the existence of an x ā¬ R with a * x = c
This follows
b+c = a*k + a*x
b+c = a*(k + x)
This follows a ā¤ (b+c), for (k+x) ā¬ R
Thank you!
The opposite is assumed, i.e. a | (b+c).
According to the first condition, a | b applies, it can be concluded directly with the assumption that:
a | (b+c) – b), i.e. a | c,
in opposition to the second condition.