Question

There is a natural relationship between sets and bit strings which is called the characteristic vector for a set. We'll look only at subsets of the universe U = {0, … , n-1} for some n, but the concept can be generalised to arbitrary sets. For a set S C U, the characteristic vector is denoted by xs and is an n-bit string where bit j is 1 if and only if j E S. For example, with n 4 and S1,3) we have xs 1010. a) What are xo and x? (1 mark) b) Given xs and xT, what is the characteristic vector of SnT? (1 mark) c) Suppose you are given xs and xT. In terms of S and T, what set is xs Ixr the characteristic vector for? (1 mark)

Answer 1

(a) X_phi = 0 0 0 0 0 - - - - - 0_n times Therefore no i elementof phi, each i^th bit is zero x_u = 1 1 1 1 1 - - - - - 1_n times every i elementof u, every i^th bit is one. (b) given x_s and x_T, the characteristic vector S Intersection T = {i: ies and X elementof T} if i^th - bit is 1 for both X_S and X_T Therefore X_S Intersection T, the i^th - bit is 1 iff i^th bit of X_S and X_T is 1 i^th - bit is 0 otherwise Therefore X_S Intersection T = X_S X_T (where product is taken bitwise) X_S/X_T = OR bit operation the set associated to it S Union T