2.∑gCm(∑) 3. Taut CCn(∑)=Cm(Cn(∑) 4.∑1∑2→M(∑2)sM(∑1) 5.Cn(∑)={|(o)= T for all v∈M(∑)} 6.σ∈Cm({σ1,,on})兮σ1→(02…→(an→a)…)∈Taat Proof. Proof of all except the property 6 just follows the definition of consequence. And you also need apply the techniques proving two sets which are equal Theorem11. For any propositions e,v,∑∪{}y分∑Fv→ yp holds Proof. Prove by the definition of consequence When we consider=, v which satisfy 2 are divided into two parts, V1(y)=T and v2() F. Then we investigate whether V satisfies y-p Conversely, v which makes v false are discarded. Because they are not taken into consider ation to satisfy∑U{} 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)分(B)→(=A) (b)A∧(BVC)分(A∧B)V(AAC 2. Prove or refute each of the following assertions (a) If either∑haor∑hB,then∑(aVB) (b)If∑(a∧B), then both∑aand∑hB 3. Prove the following assertio (a)Cm(∑)=Cm(Cn(∑). (b)∑1C∑→M(∑2)cM(∑1) (c)Cn(∑)={(0)= T for all k∈M(∑)} (d)a∈Cn({1,…,an})兮σ1→(2…→(n→a)…)∈Tat. 4. Suppose we have two assertions, where a and B both are propositions and 2 is a set of propositions2. 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. Proof. Proof of all except the property 6 just follows the definition of consequence. And you also need apply the techniques proving two sets which are equal. Theorem 11. For any propositions φ, ψ, Σ ∪ {ψ} |= φ ⇔ Σ |= ψ → φ holds. Proof. Prove by the definition of consequence. When we consider ⇒, V which satisfy Σ are divided into two parts, V1(ψ) = T and V2(ψ) = F. Then we investigate whether V satisfies ψ → φ. Conversely, V which makes ψ false are discarded. Because they are not taken into consider￾ation to satisfy Σ ∪ {ψ}. 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)}. (d) σ ∈ Cn({σ1, . . . , σn}) ⇔ σ1 → (σ2 . . . → (σn → σ). . .) ∈ T aut. 4. Suppose we have two assertions, where α and β both are propositions and Σ is a set of propositions: 5