Section 2.2: Problem 8 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
Assume that is a set of sentences such that for any sentence , either or . Assume that is a model of . Show that for any sentence , we have iff .
By assumption, either every model of is a model of or every model of is a model of . Then, using the fact that no can be both a model of and a model of , we conclude that every model of is a model of ( ) iff any model of is a model of ( ). Note that does not actually require the assumption.