3. (10 marks) Consider an ADT consisting of a set S of distinct integers and the following operations: INSERT(S,x): Insert the element x into the set S. This operation has no effect if x is already in S. DELETE(S,x): Delete the element x from the set S. This operation has no effect if r is not in S QUERY(S,x): Return true if x is in S and return false if x is not in S. CLOSEST-PAIR(S): Return two integers in S which are closest together in value. In other words, if CLOSEST-PAIR(S) returns the integers a and b, then they must satisfy the condition It is an error if S contains fewer than two elements. Describe a data structure to implement this ADT. All operations must be performed in O(log n) time, where n - |S|. Describe any new information that will be stored Justify why your algorithms are correct and why they achieve the required time bound