Description:

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Derive in details the complexity of the following code fragments in terms

of the Big$-$O notation: $($n is a very large number in all questions$)$

$1)$

for $($i $=1$; i $<=$ n; i$++)$

for $($j $=1$ j $<=$ n; j$++)\{$

x $=2*$ y;

z $=$ x$--$;

$\}$

$2)$

for $($i $=1$; i $<=$ n; i$++)\{$

j $=$ n;

while $($j $>=1)\{$

x$++$;

j $=$ j $/2$;

$\}$

$\}$

$3)$

i $=1$;

while $($i $!=$ n$)$

i$++$;

$4)$

for $($i $=1$ i $<=$ n; i$++)$

if $($i $==1||$ i $==$ n $||$ i $==3)$

for $($j $=1$ j $<=$ n; j$++)$

x$++$;

$5)$ Provide the corresponding big$-$O notation for the following functions:

n$^2+3$n $-7$

$5$n$^3-3$n$^2+8$n $-2$

lg$($n$)-2+3$n

nlg$($n$)-$ n $+3$