I'm just going through some logic lecture notes I found online. I want to verify my answers to Exercise 1.5.9 (page 30).
For a propositional theory $T$, denote by $Thm(T)$ the set of formulas provable in $T$. Decide which of the following hold: (I give my answers)
(a) $T \subseteq Thm(T), \quad $ true
(b) $Thm(Thm(T)) = Thm(T), \quad $ true
(c) $S \subseteq T$ if and only if $Thm(S) \subseteq Thm(T), \quad $ false
(d) $ S \subseteq Thm(T) $ iff $Thm(S) \subseteq Thm(T), \quad $ true
(e) $Thm(S \cup T) = Thm(S) \cup Thm(T), \quad $ false
(f) $Thm(S \cup T) = Thm(S \cup Thm(T)) = Thm(Thm(S) \cup Thm(T)), \quad $ true
(g) If $T_n \subseteq T_{n+1}$ for every $n \in N$, then $Thm(\cup T_n) = \cup Thm(T_n), \quad $ true
(h) If $T_n$ is a directed system, then $ Thm(\cup T_n) = \cup Thm(T_n) \quad $ I actaully don't know what they mean by directed system. Anyone knows? :)
Thank you!!!