Questions on runtime analysis properties

Am I wondering if the follow are true.

If f(n) is开发者_如何学Python O(g(n)) and f(n) is also Ω(g(n)) that means f(n) is also big Θ(g(n)) right? Also if either of the 2 above are false, then f(n) is not big Θ(g(n))?

If f(n) is O(g(n)) and f(n) is also Ω(g(n)) that means f(n) is also big Θ(g(n)) right?

Yes. That is the definition of big theta.

Also if either of the 2 above are false, then f(n) is not big Θ(g(n))?

Yes. It is a bijection.

if we know f1(n) is Θ(g(n)) and f2 (n) is Θ(g(n)), does that mean f1 (n) + f2 (n) is Θ(g(n)). Why?

Because f1(n) is approximately c1*g(n) and f2 is approximately c2*g(n), so f1(n)+f2(n) is approximately (c1+c2)*g(n), and so any linear combination will preserve that relationship.

Since big-omega is lower bound, big-O is upper bound, and big-theta is upper and lower bound, it stands to reason yes.