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Section 2.2: Problem 7 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
Restate the definition of “ satisfies with ” in the way described on page 84. That is, define by recursion a function such that satisfies with iff .
For an atomic formula (where can also be the equality relation) we define . Note, that for any atomic formula , iff .
We further define (i.e., by induction, iff iff iff ), (i.e., by induction, iff or iff or iff ), and (i.e., by induction, iff for every , iff for every , iff ).