Consider the set F = {1, −1, i, −i} with an operation ✕ defined by the table. ✕ 1 −1 i −i 1 1 −1 i −i −1 −1 1 −i i i i −i −1 1 −i −i i 1 −1 a ✕ b means find the entry in row a and column b; for example, −1 ✕ (−i) = i (the entry in row −1 and column −i). Find each of the following. (a)    −1 ✕ i     (b)    i ✕ i     (c)    −i ✕ i     (d)    −i ✕ 1     (e)    1 ✕ i     (f)    −i ✕ −i

1. Write an algorithm to determine whether a given element x belongs to a set S := {s1, . . . , sn}.

Let S be a set of n distinct positive integers, where n is odd.  The median of S is the (n+1)/2th smallest number of this set - i.e., the number in the middle of the set once the set is sorted. For example, if S = [70, 10, 20, 60, 30], then the sorted set is [10, 20, 30, 60, 70], from which we see that the median is 30. There is an obvious linearithmic algorithm to determine the median of S.  First you sort the set, which takes O(n log n) time.  Then you output the value of S[(n+1)/2}, which takes O(1) time.  But is this the best we can do? Here is a bold claim: there is an O(n) algorithm to determine the median of set S. Determine whether the above claim is TRUE or FALSE.