Part 1 ans:
Let us say here "n" represent some function say h(n) so that we can more intuitively define the function,
Now, if f(n) = Ω(n) means in ours terms we can say f(n) = Ω.h(n) hence similarly g(n) = Ο.h(n)
Then according to the definition
f(n) = Ω.h(n) if
f(n) ≥ c.h(n) and -----------------(1)
g(n) = O.h(n) if
g(n) ≤ c.h(n) -----------------(2)
Now, g(n) ≤ c.h(n) ≤ f(n)
case 1: IF f(n), g(n) and c.h(n) are equal then,
f(n) * g(n) = [c.h(n)]2
case 2: In all cases f(n) is greater than g(n) according to the definition (1) & (2)
Notion : f(n) > g(n)
or c1.h(n) > c2.h(n) so as due to the const. "c1" f(n) becomes larger than g(n).
Hence in this case: f(n) * g(n) = c1.c2.[h(n)]2.
Part 2 ans: f(n)/g(n) always greater than or equal to 1.
Part 3 ans: f(n)-g(n) is always positive.