$|f(n)-g(n)|\in \mathcal{O}(t(n)) $ And $f(n)+g(n)\in \Omega(t(n))$,Its true that $f(n)\in \Omega(t(n))$?

Question

I want to prove the following by the definition

1. $$|f(n)-g(n)|\in \mathcal{O}(t(n)) $$
2. $$f(n)+g(n)\in \Omega(t(n))$$

Is it true that $f(n)\in \Omega(t(n))$?
What I tried is just think about functions, it seems true, but I don't know how to prove it in formal way.
For Example:

• $f(n)=n^6$
• $g(n)=n^4$
• $t(n)=n^6$

Answer 1

No, this is not true.. Just turn around your example.. i.e. $f(n) = n^4$, $g(n) = n^6$, $t(n) = n^6$. Then both your given conditions are true, but $f(n) \notin \Omega(t(n))$.