Consider the following satisfiability problem: Fred, Daphne, Norville, Velma, an

Question

Consider the following satisfiability problem: Fred, Daphne, Norville, Velma, and Scooby are going to a movie, and they find a set of five seats for them all in a row. Some of the people in the group do not like each other, while others are good friends. In particular,

1. Norville and Velma want to sit next to each other.

2. Daphne wants to sit next to Scooby and either Fred or Norville.

3. Fred refuses to sit next to Norville or Scooby.

4. Fred wants to sit next to Daphne if and only if Daphne sits next to neither Norville nor Scooby.

5. Fred does not want to sit on the end.

6. Everybody has to sit next to at least one other person.

Let N (x, y) represent the propositional function “x and y are sitting next to one another”, where the domain for x and y is the set {F, D, N, V, S}. You may NOT define any other propositional functions for this problem.

(a) Translate each of the group’s seating requirements 1-6 from English into a proposition.

(b) Are the group’s seating requirements satisfiable? If they are, provide a seating arrangement that satisfies the requirements. If they are not, provide a concisely written argument explaining why not.

Explanation / Answer

1. N(N,V)
2. N(D,S) and (N(D,F) or N(D,N))
3. not N(F,N) or not N(F,S)
4. not N(D,N) and not N(D,S) ---> N(F,D)
5. There exist x , There exist y( N(x,F) and N(y,F))
6. For all x, there exist y (N(x,y))

A possible start can be
   D S
next step can be
V N D S

The arrangement is not possible.Dapahne has to be with Scooby. Fred choice regarding Daphane can't be fullfilled because these are contradictory.
   

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