Section 1.2: Problem 1 Solution

Working problems is a crucial part of learning mathematics. No one can learn... merely by poring over the definitions, theorems, and examples that are worked out in the text. One must work part of it out for oneself. To provide that opportunity is the purpose of the exercises.
James R. Munkres
Show that neither of the following two formulas tautologically implies the other:
Suggestion: Only two truth assignments are needed, not eight.
We need two assignments such that one of the two formulas is true and the other is false, and vice versa. If all three of , , are false, then the first expression is evaluated to be false, while the second to be true. On the other hand, if , , then the first is true, while the second is false.