Which of the following 5 sentences are true?

a) It is not the case that 2 consecutive sentences are both false.
b) There are fewer false than true sentences.
c) It is not the case that 3 consecutive sentences are all false.
d) It is not the case that 2 consecutive sentences are both true.
e) There are exactly 3 false sentences.

The answer is that D is the only statement that is true. Here is the solution.

Part of the basis of my solution is the fact that if you have 3 apples in a row, then you definitely have 2 apples in a row. Also, there aren’t 2 apples in a row anywhere, then there aren’t 3 apples in a row anywhere as well. However, you could have 2 apples in a row and not 3 apples in a row.

Why do I bring this up? Because of statements A and C. If A is true, then it follows that C must be true. However, if C is false, then A must be false as well. But these are not the only two cases that could exist. The last one is that C could be true, and A could be false. So if A then C. If !C then !A. Or !A and C is ok.

Well, I didn’t reduce the problem by that much. I reduced it from 4 to 3 cases.

B and E can’t both be true. If E is true, then it follows that B is false. Contrapositively, If B is true, then E is false. So if E then !B, and if B then !E.

Well, if C is true, then D can’t be true. Why? Because if D were true, then we would have two consecutive true statements, but D being true means that there aren’t 2 consecutive true statements. Therefore, if C is true, then D is false. D being false means that there MUST be two trues in a row somewhere. Well, in this case the only way for that to happen is if B is true (A can be anything still) because C,D can’t be T,T, and neither can D,E. So we can only have A,B or B,C be T,T. Either way B must be true. Well, the only way B could be true is if at least one of A or E is true. We already said if B then !E, so E must be false meaning A must be True for at least one of A or E to be true. So we have TTTFF. But A being true contradicts the two consecutive false statements (D and E).

Therefore our initial assumption is false, C cannot be true. Well, this removed two possible cases (C, A and C, !A). Now we went from 4 to 3 to 1 possibility for A and C.

Now it must be true that C is false. This means that A must be false. This also means that somewhere there are 3 falses in a row.

If D is false, that means there must be two true statements in a row somewhere. But since we said A and C are false, and we are supposing D is false, there is not way for there to be two true statements in a row anywhere. Therefore D must be true.

Since D is true, this means that E must be false. If E were true, that would make D false, creating a contradiction.

So we have F?FTF. Remember we said there must be 3 F’s in a row, according to C being false. Also, since E is false we can’t have exactly 3 statements be false. Therefore B must be false. And this is consistent with B’s statement.

So the sequence is FFFTF, and D is the only true statement.

Posted by nikki on 2004-10-12 18:16:29