Question $1$

$10$ pts

For each of the following pairs of functions $f(n)$ and $g(n),$ decide whether we have $f(n)in(g(n)),$ or $f(n)inO(g(n)),$ or $f(n)in(g(n)).$ If $f(n)in(g(n)),$ you will not earn any point by selecting $f(n)inO(g(n))$ or $f(n)in(g(n)),$ since $f(n)in(g(n))$ tells more than each of the other two.

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$f(n)=n^{2}lnn,g(n)={\displaystyle \underset{i=1}{\overset{n}{}}}i.$

$$

$f(n)=n^{100},g(n)={1\mathrm{.}001}^n$

$$

$f(n)=(2n)!,g(n)=2n!.$

$$

$f(n)={\displaystyle \underset{i=1}{\overset{2n}{}}}{i}^2,g(n)={\displaystyle \underset{i=1}{\overset{n}{}}}(2i{)}^2$