Rule of Sums:

$$

$($

$$

$+$

$$

$)$

$=$

$$

$($

$$

$)$

$+$

$$

$($

$$

$)$

O$($f$+$g$)=$O$($f$)+$O$($g$)$

FALSE: The correct formula is

$$

$($

$$

$+$

$$

$)$

$=$

$$

$($

max

$($

$$

$,$

$$

$)$

$)$

O$($f$+$g$)=$O$($max$($f$,$g$)).$

Rule of Products:

$$

$($

$$

$$

$$

$)$

$=$

$$

$($

$$

$)$

$$

$$

$($

$$

$)$

O$($f$$g$)=$O$($f$)$O$($g$)$

FALSE: The correct formula is

$$

$($

$$

$$

$$

$)$

$=$

$$

$($

$$

$$

$$

$)$

O$($f$$g$)=$O$($f$$g$).$

Transitivity: If

$$

$=$

$$

$($

$$

$)$

g$=$O$($f$)$ and

$$

$=$

$$

$($

$$

$)$

h$=$O$($f$),$ then

$$

$=$

$$

$($

$$

$)$

g$=$O$($h$).$

TRUE: Transitivity holds.

$5$

$$

$+$

$8$

$$

$2$

$+$

$100$

$$

$3$

$=$

$$

$($

$$

$4$

$)$

$5$n$+8$n

$2$

$+100$n

$3$

$=$O$($n

$4$

$)$

TRUE: The statement is true. The expression

$5$

$$

$+$

$8$

$$

$2$

$+$

$100$

$$

$3$

$5$n$+8$n

$2$

$+100$n

$3$

is

$$

$($

$$

$4$

$)$

O$($n

$4$

$)$ because

$100$

$$

$3$

$100$n

$3$

is the dominating term.

$5$

$$

$+$

$8$

$$

$2$

$+$

$100$

$$

$3$

$=$

$$

$($

$$

$2$

log

$$

$)$

$5$n$+8$n

$2$

$+100$n

$3$

$=$O$($n

$2$

logn$)$