We can infer from the example that truth valuation of a propostion is determined by those propositions which it is based on We define the following term to guarantee the truth of a compound proposition Definition 2(Valuation). A truth valuation v is a function that assigns to each proposi- tion a a unique truth value v(a) so that its value on a compound proposition is determined in accordance with the appropriate truth tables Here, we should remember that truth valuation determines all propositions generated ac- cording to definition. Especially, when a is a propositional letter we have V(a)=A(a)for some A Generally, we have the following theorem: Theorem 3. Given a truth assignment a there is a unique truth valuation v such that V(a)=A(a) for every propositional letter a Proof. The proof can be divided into two step 1. Construct a v from A by induction on the depth of the associated formation tree 2. Prove the uniqueness of v with the same A by induction bottom-up which show us the relation between truth assignment and truth valuation We now consider a specific proposition a. There is a corollary Corollary 4. If V1 and v2 are two valuations that agree on the support of a, the finite set of propositional letters used in the construction of the proposition of the proposition a, then vi(a)=v2(a) Given a proposition, there is a case that it is always true whatever the truth valuation is. Definition 5. A proposition g of propositional logic is said to be valid if for any valuation v, v(o)=T. Such a proposition is also called a tautology Example 2. aVna is a tautology SolutionWe can infer from the example that truth valuation of a propostion is determined by those propositions which it is based on. We define the following term to guarantee the truth of a compound proposition. Definition 2 (Valuation). A truth valuation V is a function that assigns to each proposi￾tion α a unique truth value V(α) so that its value on a compound proposition is determined in accordance with the appropriate truth tables. Here, we should remember that truth valuation determines all propositions generated ac￾cording to definition. Especially, when α is a propositional letter we have V(α) = A(α) for some A. Generally, we have the following theorem: Theorem 3. Given a truth assignment A there is a unique truth valuation V such that V(α) = A(α) for every propositional letter α. Proof. The proof can be divided into two step. 1. Construct a V from A by induction on the depth of the associated formation tree. 2. Prove the uniqueness of V with the same A by induction bottom-up. which show us the relation between truth assignment and truth valuation. We now consider a specific proposition α. There is a corollary. Corollary 4. If V1 and V2 are two valuations that agree on the support of α, the finite set of propositional letters used in the construction of the proposition of the proposition α, then V1(α) = V2(α). Given a proposition, there is a case that it is always true whatever the truth valuation is. Definition 5. A proposition σ of propositional logic is said to be valid if for any valuation V, V(σ) = T. Such a proposition is also called a tautology. Example 2. α ∨ ¬α is a tautology. Solution: α ¬α α ∨ ¬α T F T F T T 2