Problem 2. (35 points) Three children- Anna, Bjrn, and Cathy-are playing in the mud. They are honest, logical, can see each other's foreheads, but cannot see their own. LetAB, and be Boolean variables that represent whether the forehead of Anna, Bjrn, and Cathy, respectively, is mudd Every child, as soon as he or she can conclude definitively that his/her forehead is muddy, will go home to wash it. There is, however, a traffic light, and children must wait for the walk sign before they go home. And, of course, pointing out that someone's forehead is muddy is against their honor system (a) An adult named David, who does not know the honor system, comes up to the children, says "At least one of you has a muddy forehead," and leaves. Write down the Boolean formula (with variables A, rB, and rc) that represents his statement. (b) When the walk sign next turns white, no one walks home. What can you conclude about rA, B, and rc (write it down as a Boolean formula)? Justify your answer. (Hint: first think about what you can conclude about xB and rc given that Anna did not walk home; then do the same for the other two pairs. c) When the walk sign next turns white for the second time, again no one walks home. What can you conclude about rA, rB, and rc (write it down as a Boolean formula)? Justify your answer. (Hint: it may, at first, help to think about only the fact that Anna didn't walk home in the previous part, and Bjrn didn't walk home in this part. Then generalize to the other cases.) (d) Explain why every child will walk home when the walk sign turns white for the third time. (e) Note that a the children saw each other's muddy foreheads, and David's initial statement seemingly told them nothing they hadn't already known. Would they have walked home without his initial statement? Why or why not