Let's analyze each statement given the condition $f(n) = O(g(n))$.

Given Condition:

• $f(n) = O(g(n))$ means that $f(n)$ grows at most as fast as $g(n)$, or that there exists a constant $C > 0$ and $n_0$ such that $f(n) \leq C \cdot g(n)$ for all $n \geq n_0$.

Statement 1: $f(n) = o\left((f(n))^2\right)$

• Small-o notation $o\left((f(n))^2\right)$ means that $f(n)$ grows strictly slower than $(f(n))^2$.
• $f(n)$ cannot be strictly less than $(f(n))^2$ unless $f(n)$ is a constant or tends to zero, which would make the left-hand side negligible compared to the right-hand side.
• In general, this relationship does not hold for arbitrary $f(n)$ (e.g., if $f(n) = n$, $n = o(n^2)$ is true, but it is not universally true for all functions).

Conclusion: This statement does not always hold.

Statement 2: $f(n) = \Omega\left((f(n))^2\right)$

• Big-Omega notation $\Omega\left((f(n))^2\right)$ means that $f(n)$ grows at least as fast as $(f(n))^2$.
• This cannot be true because $(f(n))^2$ grows faster than $f(n)$ for sufficiently large $n$, except in trivial cases like when $f(n)$ is a constant.

Conclusion: This statement does not always hold.

Statement 3: $g(n) = o\left((f(n))^2\right)$

• Small-o notation $o\left((f(n))^2\right)$ means that $g(n)$ grows strictly slower than $(f(n))^2$.
• Given that $f(n) = O(g(n))$, this implies $(f(n))^2$ is $O\left((g(n))^2\right)$. However, $g(n) = o\left((f(n))^2\right)$ suggests $g(n)$ is strictly less than $(f(n))^2$.
• There is no guarantee that this condition will always hold. For example, if $f(n) = g(n)$, then $g(n)$ cannot grow slower than $(f(n))^2$.

Conclusion: This statement does not always hold.

Statement 4: $g(n) = \Omega(g(n))$

• Big-Omega notation $\Omega(g(n))$ means that $g(n)$ grows at least as fast as $g(n)$.
• This is trivially true because any function $g(n)$ is trivially bounded below by itself.

Conclusion: This statement always holds.

Final Answer:

• The statement that will always hold is Statement 4: $g(n) = \Omega(g(n))$.

Would you like more details or have any further questions?

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Here are some related questions that could deepen your understanding:

1. What conditions must be met for a function to be in $O(g(n))$?
2. Can you provide an example where $f(n) = o(g(n))$?
3. How does $f(n) = \Theta(g(n))$ differ from $f(n) = O(g(n))$?
4. Why does statement 4 always hold regardless of $f(n)$ or $g(n)$?
5. Can two different functions $f(n)$ and $g(n)$ both be $O(n^2)$ and still differ in their growth rates?
6. What role do constants play in Big-O and Big-Omega notations?
7. How can we interpret $f(n) = o(g(n))$ in terms of algorithm efficiency?
8. What are the implications of $g(n) = \Omega(g(n))$ in algorithm analysis?

Tip: When evaluating asymptotic statements, consider the general properties of the functions involved and how the notation affects the relationships.