Thm(T) of propositional theory T questions

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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!!!




logic propositional-calculus




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  • Your answers are correct. A nonempty family $mathcal F$ of sets is directed iff, for each $A,Binmathcal F$, there is some $Cinmathcal F$ with $Acup Bsubseteq C$.
    – Andreas Blass
    Jul 24 at 13:12














up vote
1
down vote

favorite












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!!!




logic propositional-calculus




share|cite|improve this question



















  • Your answers are correct. A nonempty family $mathcal F$ of sets is directed iff, for each $A,Binmathcal F$, there is some $Cinmathcal F$ with $Acup Bsubseteq C$.
    – Andreas Blass
    Jul 24 at 13:12












up vote
1
down vote

favorite









up vote
1
down vote

favorite











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!!!




logic propositional-calculus




share|cite|improve this question











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!!!





logic propositional-calculus





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asked Jul 24 at 8:54









Dominik Teiml

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  • Your answers are correct. A nonempty family $mathcal F$ of sets is directed iff, for each $A,Binmathcal F$, there is some $Cinmathcal F$ with $Acup Bsubseteq C$.
    – Andreas Blass
    Jul 24 at 13:12
















  • Your answers are correct. A nonempty family $mathcal F$ of sets is directed iff, for each $A,Binmathcal F$, there is some $Cinmathcal F$ with $Acup Bsubseteq C$.
    – Andreas Blass
    Jul 24 at 13:12















Your answers are correct. A nonempty family $mathcal F$ of sets is directed iff, for each $A,Binmathcal F$, there is some $Cinmathcal F$ with $Acup Bsubseteq C$.
– Andreas Blass
Jul 24 at 13:12




Your answers are correct. A nonempty family $mathcal F$ of sets is directed iff, for each $A,Binmathcal F$, there is some $Cinmathcal F$ with $Acup Bsubseteq C$.
– Andreas Blass
Jul 24 at 13:12















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