Prove Theorems 2.50-2.54 on logical consequence. Theorem 2.50 UA if and only if A; Ai ...

 

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Prove Theorems 2.50-2.54 on logical consequence. Theorem 2.50 UA if and only if A; Ai → A. Definition 2.511 A; is an abbreviation for A₁ ^ ….. ^ An. The notation ; is used if the bounds are obvious from the context or if the set of formulas is infinite. A similar notation Vis used for disjunction. Example 2.52 From Example 2.49, {p,¬q} = (pvr) ^ (¬q v¬r), so by Theo- rem 2.50,= (p ^¬q) → (pvr) ^ (¬q v¬r). The proof of Theorem 2.50, as well as the proofs of the following two theorems are left as exercises. Theorem 2.53 If U = A then UU{B} = A for any formula B. Theorem 2.54 If U = A and B is valid then U − {B} | A. Prove Theorems 2.50-2.54 on logical consequence. Theorem 2.50 UA if and only if A; Ai → A. Definition 2.511 A; is an abbreviation for A₁ ^ ….. ^ An. The notation ; is used if the bounds are obvious from the context or if the set of formulas is infinite. A similar notation Vis used for disjunction. Example 2.52 From Example 2.49, {p,¬q} = (pvr) ^ (¬q v¬r), so by Theo- rem 2.50,= (p ^¬q) → (pvr) ^ (¬q v¬r). The proof of Theorem 2.50, as well as the proofs of the following two theorems are left as exercises. Theorem 2.53 If U = A then UU{B} = A for any formula B. Theorem 2.54 If U = A and B is valid then U − {B} | A.

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Theorem 250 UAVVE V if v U then v A And all A U VU is equ... View the full answer