Let f(n) and g(n) be asymptotically positive functions. Prove or disprove each of the following conjectures. Note that to disprove a conjecture you only have to produce a single counterexample.

   1. f(n) = O(g(n)) implies g(n) = O(f(n)).
   2. f(n) + g(n) = O(min(f(n),g(n))).
   3. f(n) = O(g(n)) implies 2f(n) = O(2g(n)).
   4. f(n) = O(f(n/2)).

All of these conjectures are false.

   1. Counterexample: f(n) = 1 and g(n) = n. It's clear that for any c, n>c*1 for n large enough.
   2. Counterexample: f(n) = 1 and g(n) = n. The conjecture would imply n = O(1).   Note that the similar conjecture: f(n)+g(n) = O(max(f(n),g(n))) is true.
   3. Counterexample: f(n) = 2n and g(n) = n. It's clear that for any c, 4n > c2n for n large enough.
   4. Counterexample: f(n) = 2n.  It's clear that for any c, 4n > c2n for n large enough.