[Math] The $n$th statement in a list of $100$ statements is “Exactly $n$ of the statements in this list are false”.

discrete mathematicslogicpropositional-calculus

I was going through "Discrete Mathematics and its Applications – 7th edition" by Konneth H Rosen when I stumbled across the following question:-

The $n$th statement in a list of $100$ statements is "Exactly $n$ of the statements in this list are false".

  1. What conclusion can you draw from the statement?

  2. Answer (1) if the $nth$ statement is "At least $n$ of the statement in this list are false".

  3. Answer (2) assuming that the list contains $99$ questions.

How to conclude from the given data? If $n$ statements are false is the $n$th statement, then maybe the first statement states that exactly $1$ statement is false. This is possible because maybe some statement from the remaining $99$ statements can be false.

But, the answer given is "99th statement is true and the rest are false". How was this concluded?

Also, for part (2) and (3), how to proceed?

Best Answer

Your list is:

  • Exactly 1 statement in this list is false.
  • Exactly 2 statements in this list are false.
  • $\vdots$
  • Exactly $n$ statements in this list are false.
  • $\vdots$
  • Exactly 100 statements in this list are false.

Consider: Can all the statements be false? Can more than one statement be true?

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