[Problem 5] Consider a 1-dimensional integer P = {A is an integer array with n elements array A, with random entries, and a decision problem integer z EA.} Prove that problem P is in NP.

Problem 2: Finding an element.(i) Let n be an even positive integer. Let A be an array containing n not necessarydifferent integers. Elements in A are not necessary sorted. Describe an algorithmwhich in time O(n) tests whether there exists an element x in A that appears in Aat least n/2 times.(ii) Let S be a set containing n integers. An element x ∈ S is an approximate median ofS, if|{y ∈ S | y < x}| ≥ n/4and |{y ∈ S | y > x}| ≥ n/4Given an array A containing n different integers, not necessary sorted, describe analgorithm which in time O(n) finds all approximate medians of the set {A[i] | i =1, 2, . . . , n}.

(e) Give a simple recursive definition for the function sos(A), which gives the length of the shortest oscillating supersequence of an arbitrary array A of integers.

b A problem Q is in P and there is a polynomial-time reduction from Q to Q'. What can we say about Q'? Is Q' € P? Is Q' € NP? c Let Q be a problem defined as follows: Input: a set of numbers A = a1, a2, ..., an and a number x Output: 1 if and only if there are two values ai, ak € A such that a; +ak = x. Is Q in NP? Is Q in P? Explain your answer.