2. Let B(x) = "-10 < x < 10". Find truth set for x € D, where D = Zt is the set of all positive integers.

NE NFA = {N| N is an NFA and L(N) = Ø}. How to show it is in NP

Let x and y be integers such that x = 3 (mod 10) and y = 5 (mod 10). Find the integer z such that 97x + 3y³ z (mod 10) and 0 ≤ z ≤9.

Long chain of friends: You are given a list of people, and statements of the form “x knows y”. You are asked to find, is there a chain of k distinct people, such as x1 knows x2, x2 knows x3, and xk-1 knows xk. Prove that this problem is NP-complete by using one of the known NP-complete problems (CLIQUE, 3-SAT, Hamiltonian Path, Hamiltonian Cycle, Independent Set, etc.)

| Consider the elliptic curve group based on the equation y? = x' + ax + b mod p where a = 4, b = 1, and p = 7. This curve contains the point P = (0, 1). The order of this elliptic curve group is the prime number 5, and therefore we can be sure that P is a primitive element. Another element in this group is Q = (0,6). The index of Q with respect to P is the least positive integer d such that Q = dP. What is d, the index of Q?

Consider the problem of making change for n cents using the fewest number of coins. Assume that we live in a country where coins come in k dierent denominations c1, c2, . . . , ck, such that the coin values are positive integers, k ≥ 1, and c1 = 1, i.e., there are pennies, so there is a solution for every value of n. For example, in case of the US coins, k = 4, c1 = 1, c2 = 5, c3 = 10, c4 = 25, i.e., there are pennies, nickels, dimes, and quarters. To give optimal change in the US for n cents, it is sufficient to pick as many quarters as possible, then as many dimes as possible, then as many nickels as possible, and nally give the rest in pennies.   Design a bottom-up (non-recursive) O(nk)-time algorithm that makes change for any set of k different coin denominations. Write down the pseudocode and analyze its running time. Argue why your choice of the array and the order in which you fill in the values is the correct one. Notice how it is a lot easier to analyze the running time of…

Let f (n) and g(n) be functions with domain {1, 2, 3, . . .}. Prove the following: If f(n) = O(g(n)), then g(n) = Ω(f(n)).

3. Let p(n) = Σª a¡n², where a¿ > 0, be a degree-d polynomial in n, and let k be a constant. Prove the following properties. if k ≥ d, then p(n) = 0(nk); if k ≤d, then p(n) = N(nk); if k = d, then p(n) = 0(nk); if k > d, then p(n) = o(n²); if k < d, then p(n) = w(nk); a. b. C. d. e.

In this group of problems, you are given the predicate P(x), where the domain of x is the set of natural numbers.

Let A = {(N1, N2) | N1 and N2 are NFAS and L(N1)N L(N2) = Ø}. Show that A is decidable.