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Section 2.2: Problem 5 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 the formula (where is a one-place function symbol and is a two-place predicate symbol) is valid.
Let be a structure and . Then, iff or iff or or . So, assume that does not hold, i.e. . Then , implying that iff iff iff , i.e. if then at least one of the other two conditions holds. So, we have that for every and , .