# Selecting the true statements for properties of $\sim$, $O$ and $o$

The restrictions on the function in the statements are the following:

• The functions are positive
• They are monotonically increasing
• $$f,g: [1,\infty) \to [1, \infty)$$
• They go to $$+\infty$$ when $$x \to +\infty$$
• All of the statements below imply $$x \to +\infty$$

The statements are:

1. If $$f(x) \sim g(x)$$, then $$g(x) \sim f(x)$$
2. If $$f(x) = o(g(x))$$, then $$g(x)=o(f(x))$$
3. If $$f(x) = O(g(x))$$, then $$g(x)=O(f(x))$$
4. If $$f(x) \sim g(x)$$, then $$(f(x))^2 \sim (g(x))^2$$
5. If $$f(x) \sim g(x)$$, then $$\ln (f(x)) \sim \ln (g(x))$$
6. If $$f(x) = O(g(x))$$, then $$\ln(f(x)) = O(\ln(g(x))$$
7. If $$f(x) \sim g(x)$$, then $$2^{f(x)} \sim 2^{g(x)}$$
8. If $$f(x) = o(g(x))$$, then $$2^{f(x)} = o(2^{g(x)})$$
9. If $$f(x) = O(g(x))$$, then $$2^{f(x)} = O(2^{g(x)})$$

The options that are given in this exercise change all the time, so here are the options that I selected to be correct from my previous answer that are not in the above list:

1. If $$f(x)=o(g(x))$$, then $$(f(x))^2 = o((g(x))^2)$$
2. If $$f(x) = O(g(x))$$, then $$(f(x))^2 = O((g(x))^2)$$

My initial answer, which was incorrect, is that the following set has the correct statements: $$\{1, 4, 10, 11, 5, 8, 9\}$$. It also, looks like for the current options, $$2$$ and $$3$$ are correct, as they are equivalent to $$10$$ and $$11$$. So really, just the options $$6$$ and $$7$$ are false in my view. But that is an incorrect answer.

For $$7$$ and $$6$$ I have found the counterexamples to be: $$x$$ and $$x+1$$, $$2x$$ and $$x$$, respectively.

Always assume $$x \to +\infty$$:
(6) is true: $$\frac{f}g < M$$ means $$\ln f - \ln g < \ln M$$, so $$\frac{\ln f}{\ln g} < \frac{M}{\ln g} +1 < 2$$ once $$\ln g$$ gets big enough.
(9) is not true: Take $$f(x)=x+\log_2 x, g(x) = x$$, then you get $$\frac{2^f}{2^g}=x$$.