This is another exercise problem (Exercise 3 of Section 2.4) of Enderton's "A mathematical introduction to logic".

(a) Let be a structure and let . Define a truth assignment on the set of prime formulas by iff . Show that for any formula (prime or not), iff .

(b) Conclude that if tautologically implies , then logically implies .

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What I have tried so far is

For (a), if is a prime formula, nothing needs to be done. If is not a prime formula, two cases have to be considered, i.e., or , where and are both atomic formulas.

For (b), if for every , then by assumption. Then I need to show if , then . I am stuck here.

Any help will be appreciated.