Let G be a set defined as follows: if z is a propositional variable, then...

 

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Let G be a set defined as follows: • if z is a propositional variable, then z € G: • if f1, f2 € G, then f₁ € G. (fi V f2) € G. and (f₁ ^ f2) = G; • nothing else belongs to G. Use structural induction to prove that for every f € G, there exists f' € G such that f and f' are logically equivalent, and f' does not contain a symbol. (Recall that propositional formulas fi and f2 are logically equivalent if fi and f2 evaluate to the same value, no matter how their variables are set.) Let G be a set defined as follows: • if z is a propositional variable, then z € G: • if f1, f2 € G, then f₁ € G. (fi V f2) € G. and (f₁ ^ f2) = G; • nothing else belongs to G. Use structural induction to prove that for every f € G, there exists f' € G such that f and f' are logically equivalent, and f' does not contain a symbol. (Recall that propositional formulas fi and f2 are logically equivalent if fi and f2 evaluate to the same value, no matter how their variables are set.)

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