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.

Firstly, we must establish that statement (c) has to be false. If it were true...

If (a) is true, (b) and (d) would have to be true, and (d) would form a contradiction, since (a) and (b) are true.

If (a) is false, (b) must also be false. Otherwise, (d) and (e) are false, and (e) makes (b) false, making (d) true.  But if (b) is false, either (d) or (e) must be false. If (d) is false, it is impossible. If (e) is false, it forces (d) to be true, which is impossible.

Therefore, (a) can neither be true nor false, so (c) must be false.

We assume (a)(b)&(c) are false, but then (d) could not be true, since (d) and (e) are consecutive and they are both true. (e) can be false if (d) is true, and I shall discuss this later. If (d) were false, (e) must be false, and all the statements are false, making (d) true!

If (b)(c)&(d) are false, then (d) would be forced to say that 2 consecutive sentences are true, and that is not possible with (a)&(e) left.

If (c)(d)&(e) are false, one more statement has to be false, making (d) false.

Therefore, the only possible answer is that (d) is the only true statement. (a) is false, since both (a) and (b) are false. (b) is false, since there is only one true statement (d). (c) is false, since (a), (b) and (c) are all false. (d) is true, since there is only one true sentence. (e) is false, since four sentences are false, not three.

Regards,
wcaexqdz

Posted by wcaexqdz on 2004-10-18 13:34:19