"Which answer in this list is the correct answer to this question?" I received this question from my mathematics professor as a leisure-time logic quiz, and although I thought I answered it right, he denied. Can someone explain the reasoning behind the correct solution?

Which answer in this list is the correct answer to this question?
  
  
*
  
*All of the below.
  
*None of the below.
  
*All of the above.
  
*One of the above.
  
*None of the above.
  
*None of the above.
  

I thought:


*

*$2$ and $3$ contradict so $1$ cannot be true.

*$2$ denies $3$ but $3$ affirms $2,$ so $3$ cannot be true

*$2$ denies $4,$ but as $1$ and $3$ are proven to be false, $4$ cannot be true.

*$6$ denies $5$ but not vice versa, so $5$ cannot be true.


at this point only $2$ and $6$ are left to be considered. I thought choosing $2$ would not deny $1$ (and it can't be all of the below and none of the below) hence I thought the answer is $6.$
I don't know the correct answer to the question. Thanks!

You can use propositional logic to formalize the problem, then satisfying assignments help to find the solutions.
Let $a,b,c,d,e,f$ represent the six sentences, respectively.


*

*$a\leftrightarrow b\land c\land d\land e\land f$

*$b\leftrightarrow \neg c\land \neg d\land \neg e\land \neg f$

*$c\leftrightarrow a\land b$

*$d\leftrightarrow (a\land \neg b\land \neg c)\lor(\neg a\land b\land \neg c)\lor (\neg a\land \neg b\land c)$

*$e\leftrightarrow \neg a\land \neg b\land \neg c\land \neg d$

*$f\leftrightarrow \neg a\land \neg b\land \neg c\land \neg d\land \neg e$


Assuming there is at least one solution: 7. $a\lor b\lor c \lor d\lor e\lor f$
The only satisfying truth assignment is the one which sets $a,b,c,d,f$ to false and set $e$ true. So choice 5 is the solution.
