 Quote by blastoise
It's false, because you when you say if and only if it is the same things as
If-And-Only-If Proofs
Often, a statement we need to prove is of the form
\X if and only if Y ." We are then required to do
two things:
1. Prove the if-part: Assume Y and prove X.
2. Prove the only-if-part: Assume X, prove Y .
taken from http://infolab.stanford.edu/~ullman/...es/slides1.pdf
Did 1.
But, number 2 is
Assume n is divisible by b and n is divisible by a if n is divisible by ab
Choose n = 8, b = 2 a = 3
n is divisible by b and n is divisible by a but n is not divisible by ab
so it's false
thx norwegian i see what you mean |
8 is not divisible by 3.
let's pick a better example, where a and b have "some factor in common".
so suppose a = 6, and b = 15, and n = 30. then a|n (because 30 = 6*5), and b|n (because 30 = 15*2), but it's pretty obvious ab = 90 does NOT divide 30 (for one, it's bigger).
in general, you only know that n is divisible by the least common multiple of a and b. in our example above, lcm(6,15) = 30, and indeed 30 divides 30.