Represent the statement in symbols as \((P \imp Q) \vee (Q \imp R)\text{,}\) where \(P\) is the statement “you get more doubles than any other player,” \(Q\) is the statement “you will lose,” and \(R\) is the statement “you must have bought the most properties.” Now make a truth table.
The truth table needs to contain 8 rows in order to account for every possible combination of truth and falsity among the three statements. Here is the full truth table:
| \(P\) | \(Q\) | \(R\) | \(P \imp Q\) | \(Q \imp R\) | \((P \imp Q) \vee (Q \imp R)\) |
| T | T | T | T | T | T |
| T | T | F | T | F | T |
| T | F | T | F | T | T |
| T | F | F | F | T | T |
| F | T | T | T | T | T |
| F | T | F | T | F | T |
| F | F | T | T | T | T |
| F | F | F | T | T | T |
The first three columns are simply a systematic listing of all possible combinations of T and F for the three statements (do you see how you would list the 16 possible combinations for four statements?). The next two columns are determined by the values of \(P\text{,}\) \(Q\text{,}\) and \(R\) and the definition of implication. Then, the last column is determined by the values in the previous two columns and the definition of \(\vee\text{.}\) It is this final column we care about.
Notice that in each of the eight possible cases, the statement in question is true. So our statement about monopoly is true (regardless of how many properties you own, how many doubles you roll, or whether you win or lose).