1. SECand 2.if∈Tisn- ary and s1,…,sn∈C, then f(81,…,sn)∈C ith these concepts, we have an Theorem on the property of inductive definition Theorem 8. Suppose that s is closed under the operations of T, there is a smallest closure C such chat scc Proof. Given a set D, sCD and D is closed under the operations of T. Consider the set C=nDISCDAD is closed under the operations of Th It is obvious that S CC. And C is the smallest set according to definition of C As a direct application of Theorem 8, we have the following Theorem about well-defined Definition Theorem 9. Ifs is a set of well defined propositions and is closed under all five connectives, then s is the set of all well defined propositions Exercises 1. Show that there are no well-defined propositions of length 2, 3, or 6, but that any other positive length is possible 2. Given an example such that(aAB)=(rns)but afr, where a and B are well-defined and r and d are two expressions. 3. Draw formation tree of the following propositions (a)(AVB)→(C)∧D) (b)((A)VB)∧(A(B)) 4. Let a be a well-defined proposition; let c be the number of places at which binary connective symbols(A,V,t, +)occur in a; let s be the number of places at which proposition letters occur in a. Show by using the induction principle that s=c+l 5. Determine the Polish notation of proposition or vice versa (a)(AVB)→(=C)∧D) (b)(A)B)∧(A(=B) (c)→∧-AB分C=D 6. Design a algorithm which can translate Polish notation of proposition into well defined propo- sition without stack or with at most one 7. Design a algorithm to translate a well-defined proposition into a Polish notation form1. S ⊆ C and 2. if f ∈ T is n-ary and s1, . . . , sn ∈ C, then f(s1, . . . , sn) ∈ C. With these concepts, we have an Theorem on the property of inductive definition. Theorem 8. Suppose that S is closed under the operations of T, there is a smallest closure C such that S ⊆ C. Proof. Given a set D, S ⊆ D and D is closed under the operations of T. Consider the set C = ∩{D|S ⊆ D ∧ D is closed under the operations of T}. It is obvious that S ⊆ C. And C is the smallest set according to definition of C. As a direct application of Theorem 8, we have the following Theorem about well-defined Definition: Theorem 9. If S is a set of well defined propositions and is closed under all five connectives, then S is the set of all well defined propositions Exercises 1. Show that there are no well-defined propositions of length 2, 3, or 6, but that any other positive length is possible. 2. Given an example such that (α ∧ β) = (γ ∧ δ) but α 6= γ, where α and β are well-defined and γ and δ are two expressions. 3. Draw formation tree of the following propositions: (a) ((A ∨ B) → ((¬C) ∧ D)). (b) (((¬A) ∨ B) ∧ (A ∨ (¬B))). 4. Let α be a well-defined proposition; let c be the number of places at which binary connective symbols(∧, ∨, →, ↔ ) occur in α; let s be the number of places at which proposition letters occur in α. Show by using the induction principle that s = c + 1. 5. Determine the Polish notation of proposition or vice versa. (a) ((A ∨ B) → ((¬C) ∧ D)). (b) (((¬A) ∨ B) ∧ (A ∨ (¬B))). (c) → ∧¬AB ↔ C¬D. 6. Design a algorithm which can translate Polish notation of proposition into well defined propo￾sition without stack or with at most one. 7. Design a algorithm to translate a well-defined proposition into a Polish notation form. 6