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Section 1.2: Problem 14 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
Let be the set of all sentence symbols, and assume that is a truth assignment. Show there is at most one extension meeting conditions 0–5 listed at the beginning of this section. (Suppose that and are both such extensions. Use the induction principle to show that .)
As before, we assume is the set of wffs on which the two extensions agree. Condition 0 ensures that . Now, if , then conditions 1-5 ensure that . Hence, by the induction principle, .