CSCI 2670 HW 1 Solutions 8/31/2004 0.4 A×B contains ab elements. For every element in A, there are b elements in A×B. 0.5 The power set of C contains 2c elements. Let C = {x1, x2, …, xc} … i.e., each element of c is associated with a unique positive integer between 1 and c. Represent each subset of C by a binary number containing c bits – the ith bit will be a 1 if and only if xi is in the subset. Clearly, each subset of c is represented by exactly one c-length binary number. Also, every such binary number represents exactly one subset of C. Therefore, the size of the power set of C is equal to the number of c-length binary numbers – namely, 2c. 0.7a “Works in the same building” is reflexive and symmetric, but not transitive. Clearly, everyone works in the same building as him- or herself. Also, if person a works in the same building as person b, then person b works in the same building as person a. However, if person b has two jobs, then it is possible for b to work in the same building as both a and c, even though a and c do not work in the same building. 0.10 In the step where you divide by (a-b), you will be dividing by zero if a = b. Therefore, the proof is only correct when a b. 1.1a q1 1.1b {q2} 1.1f No. aabb ends up at state q1, which is not an accept state. 1.1g Yes. Since the start state is a member of the set of accepting states, the empty string is accepted. 1.2 Q = {q1, q2, q3, q4} = {a, b} q1 q2 q3 q4 q0 = q1 F = {q1, q4} a q1 q3 q2 q3 b q2 q4 q1 q4 1.4b 0 0 0 1 0,1 1 1 1.4d 0,1 0,1 0,1 0 0,1 1 1.4f 0 0 1 1 0,1 1 0 1.4h 0,1 1 1 1 0 0 0 0 1.4j 0 1 0 1 1 0 1 0 0 1 1 0,1 0,1