Consider the following Algorithms, Analyse them fully

Bubble Sort

BruteForceStringMatch Algorithm

Prove $($by using the definitions of the notations involved$)$ or disprove $($by giving a specific counterexample$)$ the following assertions.

If t$($n$)$ in O$($g$($n$)),$then g$($n$)$ in $\backslash$Omega $($t$($n$)).(3)$

$\backslash$Theta $(\backslash$alpha g$($n$))=\backslash$Theta $($g$($n$)),$where $\backslash$alpha $>0.(5)$

$\backslash$Theta $($g$($n$))=$ O$($g$($n$))\backslash$cap $\backslash$Omega $($g$($n$)).(3)$

For any two nonnegative functions t$($n$)$ and g$($n$)$ defined on the set of nonnegative integers, either t$($n$)$ in O$($g$($n$)),$ or t$($n$)$ in $\backslash$Omega $($g$($n$)),$ or both.

NB: It is important to consult the relevant material before attempting to answer these questions