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Section 1.2: Problem 2 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
(a) Is a tautology?
(b) Define recursively as follows: and . For which values of is a tautology? (Part (a) corresponds to k = 2.)
(a) It can be false only if is false, but then is true, and is false, implying the whole expression is true. Contradiction, hence, it is a tautology.
(b) If , then it is obviously not a tautology. If , and is false, then is false, but then is true (regardless of the value assigned to ), is false, is true, etc. Hence, it is a tautology for even values of , except for .