MATH 0050: Logic2023-2024Exercise Set B1) Below, α, β , γ , δ are taken to be propositions.a) Use the semantic tableaux method to show that the following semantic implication holds:{α ⇒ (((¬ (β ⇒ γ)) ⇒ α) ⇒ (¬δ))} |= α ⇒ (¬δ)b) Write down a direct proof of the following syntactic implication (a proof that does not assume the validity of any theorems and does not use the Deduction Theorem):{α ⇒ (((¬ (β ⇒ γ)) ⇒ α) ⇒ (¬δ))} ⊢ α ⇒ (¬δ)2) Below, α, β , γ , δ , ϵ are taken to be propositions.a) Use the Deduction Theorem for propositional logic to show the following:{(¬β) ⇒ (¬γ) , (¬α) ⇒ ((¬β) ⇒ γ)} ⊢ (¬α) ⇒ βb) Use the Deduction Theorem for propositional logic to show the following:{α ⇒ ((β ⇒ (¬γ)) ⇒ ((¬δ) ⇒ ϵ))} ⊢ (β ⇒ (¬γ)) ⇒ ((¬δ) ⇒ (α ⇒ 代 写MATH 0050: Logic 2023-2024 Exercise Set BMatlab ϵ))3) After defining, in each case, a suitable set of predicate symbols (Π) and a suitable set of functional symbols (Ω), write down a theory (i.e. a set of sentences) that has as (normal) models (only):a) all posets containing at most 5 elements;b) all graphs in which every vertex is connected, by edges, to at least 4 other vertices.4) a) Show that the following functions are computable:(ii) f2 : N0(2) → N0 , f2 (m,n) = 5m + 3n + 4b) Show that the following functions are recursive:(i) g1 : N0 → N0 , g1 (m) = 20(ii) g2 : N0(2) → N0 , g2 (m,n) = 7mn + 20For part (b)(ii), you may assume that h : N0(2) → N0 , h(m,n) = m + n, is recursive.WX：codehelp
