7. (10+10 points) (a) We say that a set I of formulas in a language L...

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7. (10+10 points) (a) We say that a set I of formulas in a language L is complete if it is consistent and for each formula in L, exactly one of y and is in I. Show that if I is complete, then it is maximal consistent (i.e. every set properly containing I is inconsistent). (b) Prove (3.4.10) that for all I c S(P) and 4,0 € S(P), if ry, then ry[0/x] provided propositional variable x does not occur in formulas of I used in derivation of rty. 7. (10+10 points) (a) We say that a set I of formulas in a language L is complete if it is consistent and for each formula in L, exactly one of y and is in I. Show that if I is complete, then it is maximal consistent (i.e. every set properly containing I is inconsistent). (b) Prove (3.4.10) that for all I c S(P) and 4,0 € S(P), if ry, then ry[0/x] provided propositional variable x does not occur in formulas of I used in derivation of rty.