Chapter 11.2, Problem 51ES

To determine

(a)

To prove:

That for the real-valued functions $f_1$, $f_2$ and $g$, if $f_1\left(n\right)\text{ is }\Theta \left(g\left(n\right)\right)$ and $f_2\left(n\right)\text{ is }\Theta \left(g\left(n\right)\right)$ then $\left(f_1\left(n\right)+f_2\left(n\right)\right)\text{ is }\Theta \left(g\left(n\right)\right)$.

To determine

(b)

To prove:

That for the real-valued functions $f_1$, $f_2$, $g_1$ and $g_2$, if $f_1\left(n\right)\text{ is }\Theta \left(g_1\left(n\right)\right)$ and $f_2\left(n\right)\text{ is }\Theta \left(g_2\left(n\right)\right)$ then $f_1\left(n\right)f_2\left(n\right)\text{ is }\Theta \left(g_1\left(n\right)g_2\left(n\right)\right)$.

To determine

(c)

To prove:

That for the real-valued functions $f_1$, $f_2$, $g_1$ and $g_2$, if $f_1\left(n\right)\text{ is }\Theta \left(g_1\left(n\right)\right)$ and $f_2\left(n\right)\text{ is }\Theta \left(g_2\left(n\right)\right)$ then $f_1\left(n\right)+f_2\left(n\right)\text{ is }\Theta \left(g_2\left(n\right)\right)$ when $g_1\left(n\right)\le g_2\left(n\right)$.