do all the function implmentations the structs are given as:$//$ DO NOT CHANGE THE NAME OF THIS STRUCT

struct set $\{$

struct node $*$tree; $//$ DO NOT MODIFY$/$REMOVE THIS FIELD

$//$ You may add more fields here if needed

$\}$;

$//$ DO NOT CHANGE THE NAME OF THIS STRUCT

struct node $\{$

int item; $//$ DO NOT MODIFY$/$REMOVE THIS FIELD

struct node $*$left; $//$ DO NOT MODIFY$/$REMOVE THIS FIELD

struct node $*$right; $//$ DO NOT MODIFY$/$REMOVE THIS FIELD

$//$ You ma$6$y add more feields here if needed

$\}$;

$//$ You may define more structs here if needed

$////////////////////////////////////////////////////////////////////////$

$//$ Cursors

struct cursor $\{$

$//$ You may add more fields here if needed

$\}$;

$////////////////////////////////////////////////////////////////////////$ The interface is given as: $//!!!$ DO NOT MODIFY THIS FILE $!!!$

#ifndef SET$\_$H

#define SET$\_$H

#include

#include

#include

#define UNDEFINED INT$\_$MIN

typedef struct set $*$Set;

$////////////////////////////////////////////////////////////////////////$

$//$ Basic Set Operations

$/**$

$*$ Creates a new empty set

$*/$

Set SetNew$($void$)$;

$/**$

$*$ Frees all memory allocated to the set

$*/$

void SetFree$($Set s$)$;

$/**$

$*$ Inserts an item into the set

$*/$

void SetInsert$($Set s$,$ int item$)$;

$/**$

$*$ Deletes an item from the set

$*/$

void SetDelete$($Set s$,$ int item$)$;

$/**$

$*$ Returns the number of elements in the set

$*/$

int SetSize$($Set s$)$;

$/**$

$*$ Returns true if the set contains the given item, and false otherwise

$*/$

bool SetContains$($Set s$,$ int item$)$;

$/**$

$*$ Prints the elements in the set to the given file in ascending order

$*$ between curly braces, with items separated by a comma and space

$*/$

void SetPrint$($Set s$,$ FILE $*$out$)$;

$////////////////////////////////////////////////////////////////////////$

$//$ Common Set Operations

$/**$

$*$ Returns a new set representing the union of the two sets

$*/$

Set SetUnion$($Set s$1,$ Set s$2)$;

$/**$

$*$ Returns a new set representing the intersection of the two sets

$*/$

Set SetIntersection$($Set s$1,$ Set s$2)$;

$/**$

$*$ Returns true if the two sets are equal, and false otherwise

$*/$

bool SetEquals$($Set s$1,$ Set s$2)$;

$/**$

$*$ Returns true if set s$1$ is a subset of set s$2,$ and false otherwise

$*/$

bool SetSubset$($Set s$1,$ Set s$2)$;

$////////////////////////////////////////////////////////////////////////$

$//$ Index Operations

$/**$

$*$ Returns the element at the given index, or UNDEFINED if the given

$*$ index is outside the range $[0,$ n $-1]$ where n is the size of the set

$*/$

int SetAtIndex$($Set s$,$ int index$)$;

$/**$

$*$ Returns the index of the given element in the set if it exists, and

$*-1$ otherwise

$*/$

int SetIndexOf$($Set s$,$ int elem$)$;

$////////////////////////////////////////////////////////////////////////$

$//$ Cursor Operations

typedef struct cursor $*$SetCursor;

$/**$

$*$ Creates a new cursor positioned at the $*$start$*$ of the set

$*($see the spec for details$)$

$*/$

SetCursor SetCursorNew$($Set s$)$;

$/**$

$*$ Frees all memory allocated to the given cursor

$*/$

void SetCursorFree$($SetCursor cur$)$;

$/**$

$*$ Returns the element at the cursor's position, or UNDEFINED if the

$*$ cursor is positioned at the $*$start$*$ or $*$end$*$ of the set

$*/$

int SetCursorGet$($SetCursor cur$)$;

$/**$

$*$ Moves the cursor to the next greatest element, or to the end of the

$*$ set if there is no next greatest element. Does not move the cursor if

$*$ it is already at the end. Returns false if the cursor is at the end

$*$ after this operation, and true otherwise.

$*/$

bool SetCursorNext$($SetCursor cur$)$;

$/**$

$*$ Moves the cursor to the next smallest element, or to the start of the

$*$ set if there is no next smallest element. Does not move the cursor if

$*$ it is already at the start. Returns false if the cursor is at the

$*$ start after this operation, and true otherwise.

$*/$

bool SetCursorPrev$($SetCursor cur$)$;

$////////////////////////////////////////////////////////////////////////$

#endif. Implement All the following functions in c using this info. pleaee provide full code for all $7$ functions and ensure time complexity is matching what is required in the table. Give explanations and use recursion where possible without creating any extra constants or helper functions. DO ALL $6$ FUNCTIONS IN C AND PLEASE DOUBLE CHECK INTERFACE BECAUSE SOME OF THEM ARE NOT STRUCTS OR POINTERS SO COPY GIVEN INTERFACE IN CODE TO MAKE SURE. FOR E$.$G$.$ IT IS SET S NOT SET $*$S$.$