$1.$ Algorithm Analysis. In this question, you will need to analyze three algorithms and

prove the required correctness and$/$or running time properties.

$($a$)$ You are given the following algorithm which accepts as input a list of positive integers

A $=($a$1,...,$ an$)$ and we need to determine whether all elements are unique $($that is$,$

there are no repeated elements in the list$).$ If all elements are unique, the algorithm

outputs True, otherwise it will output False.

$1$: procedure CHECKUNIQUE$($A$,$ n$)$

$2$: for i $=1$ to n $1$ do

$3$: for j $=$ i $+1$ to n do

$4$: if ai $==$ aj then

$5$: return FALSE

$6$: end if

$7$: end for

$8$: end for

$9$: return TRUE

$10$: end procedure

$($i$)(3$ points$)$ State the primitive operations and the data structure$($s$)$ to be used in

this algorithm.

$($ii$)(2$ points$)$ Compute the worst$-$case running time T $($n$)$ to output the final result

using $\backslash$Theta $-$notation.