complex analysis, predicate calc

solve number 2

DEDUCTIVE CLOSURES A set of formulae is deductively closed iff A to implies that OEA Below, I denotes a set of sentences. (1) Show that if {4;} are all deductively closed sets of formulae so is nA. (2) Show that any set of sentences is contained in a smallest deductively closed set A(T). (3) Show that A(T) = {o:ro). (4) Show that if S is complete, then it is deductively closed. (5) Show that if A(T) n {0, -o} = 0, then A(T) U{} and A(T) U{-o} are both consistent. (6) Show that A(T) is the intersection of all maximal consistent containing I. (7) Define SA as all sentences in the language which are true in the structure A. Show that A(T) is the intersection of all A SO that all sentences of I are true in the structure A. DEDUCTIVE CLOSURES A set of formulae is deductively closed iff A to implies that OEA Below, I denotes a set of sentences. (1) Show that if {4;} are all deductively closed sets of formulae so is nA. (2) Show that any set of sentences is contained in a smallest deductively closed set A(T). (3) Show that A(T) = {o:ro). (4) Show that if S is complete, then it is deductively closed. (5) Show that if A(T) n {0, -o} = 0, then A(T) U{} and A(T) U{-o} are both consistent. (6) Show that A(T) is the intersection of all maximal consistent containing I. (7) Define SA as all sentences in the language which are true in the structure A. Show that A(T) is the intersection of all A SO that all sentences of I are true in the structure A