2. Let A(A)=T, A(B=T,A(C)=T 3. Let A(A)=T, A(B)=T, A(C)=F, A(D)=F, Definition 9. We say that propositions 2 is satisfiable if it has some model. Otherwise it is called invalid Proposition 10. Let 2, 21, 22 be sets of propositions. Let Cn(e) denote the set of conse quence of 2 and Taut the set of tautologies 1.∑1s∑2→Cn(1)sCn(∑2) 2.∑Cn(∑) 3. Taut CCn(∑)=Cn(Cn(∑) 4.∑1∑2→M(∑2)sM(∑1) 5.Cn(∑)={|(a)= T for all∈M∑)} 6.σ∈Cn({a1,…,on}兮a1→(02…→(n→0)…)∈Tat Theorem11. For any propositions e,v,∑∪{}y分∑Fv→ p holds. Proof. Prove by the definition of consequence With this Theorem 11, we can prove result 6 in Proposition 10 by induction Exercises 1. Check whether the following propositions are valid or not (b)A∧(BVC)分(A∧B)V(AAC) 2. Prove or refute each of the following assertions (a) If either∑haor∑hB,then∑F(aVB) (b)If∑(aAB), then both∑ F a and∑hB 3. Prove the following assertion (a)Cn(∑)=Cn(Cm(∑). (b)∑1C∑2→M(∑2)cM(∑1) (c)Cn(∑)={0|()= T for all v∈M∑)}2. Let A(A) = T, A(B) = T, A(C) = T. 3. Let A(A) = T, A(B) = T, A(C) = F, A(D) = F, . . .. Definition 9. We say that propositions Σ is satisfiable if it has some model. Otherwise it is called invalid. Proposition 10. Let Σ, Σ1, Σ2 be sets of propositions. Let Cn(Σ) denote the set of conse￾quence of Σ and T aut the set of tautologies. 1. Σ1 ⊆ Σ2 ⇒ Cn(Σ1) ⊆ Cn(Σ2). 2. Σ ⊆ Cn(Σ). 3. T aut ⊆ Cn(Σ) = Cn(Cn(Σ)). 4. Σ1 ⊆ Σ2 ⇒ M(Σ2) ⊆ M(Σ1). 5. Cn(Σ) = {σ|V(σ) = T for all V ∈ M(Σ)}. 6. σ ∈ Cn({σ1, . . . , σn}) ⇔ σ1 → (σ2 . . . → (σn → σ). . .) ∈ T aut. Theorem 11. For any propositions ϕ, ψ, Σ ∪ {ψ} |= ϕ ⇔ Σ |= ψ → ϕ holds. Proof. Prove by the definition of consequence. With this Theorem 11, we can prove result 6 in Proposition 10 by induction. Exercises 1. Check whether the following propositions are valid or not (a) (A → B) ↔ ((¬B) → (¬A)) (b) A ∧ (B ∨ C) ↔ (A ∧ B) ∨ (A ∧ C) 2. Prove or refute each of the following assertions: (a) If either Σ |= α or Σ |= β, then Σ |= (α ∨ β). (b) If Σ |= (α ∧ β), then both Σ |= α and Σ |= β. 3. Prove the following assertion: (a) Cn(Σ) = Cn(Cn(Σ)). (b) Σ1 ⊂ Σ2 ⇒ M(Σ2) ⊂ M(Σ1). (c) Cn(Σ) = {σ | V(σ) = T for all V ∈ M(Σ)}. 4