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 .
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.