Anna wrote down the following 3 statements, denoted P1, P2, P3, on a blank piece of paper Here's a problem presented in my proofs class that I cannot for the life of me figure out. Apparently the answer is P2 according to the answer book but I have no idea how Help is much appreciated, thanks in advance!

Anna wrote down the following 3 statements, denoted P1, P2, P3, on a blank piece of paper:

P1 : There is exactly one FALSE statement written on this piece of paper.

P2 : There are exactly two FALSE statements written on this piece of paper .

P3 : There are exactly three FALSE statements written on this piece of paper.

One of the above statements is TRUE. What statement is TRUE?
Hint. If exactly one of the above statements is TRUE, and there are exactly three statements in all, then exactly how many are FALSE?
You're told that one of these statement &mdash; which means precisely one of them &mdash; is TRUE. So you can simply consider all possible cases in a very straightforward manner:


maybe P1 is TRUE and the other two are FALSE;
or maybe P2 is TRUE and the other two are FALSE;
or maybe P3 is TRUE and the other two are FALSE.


When you consider these three possible cases, you'll see that only one of them works, while the other two lead to a contradiction.

EDIT. Note that we don't even need to be told that one of them is true. Even without this additional information there's only one non-contradictory way (the same way) to assign TRUE/FALSE values to these three statements.