Discrete Mathematics(II)

Discrete Mathematics(II) Yi Li Software School Fudan University March 27, 2017 Yi Li (Fudan University) Discrete Mathematics(II) March 27, 2017 1 / 28

Review Language Truth table Connectives Yi Li (Fudan University) Discrete Mathematics(II) March 27, 2017 2 / 28

Outline Formation tree Assignment, Valuation, and Deduction Yi Li (Fudan University) Discrete Mathematics(II) March 27, 2017 3 / 28

Ambiguity Example Consider the following sentences: Yi Li (Fudan University) Discrete Mathematics(II) March 27, 2017 4 / 28

Ambiguity Example Consider the following sentences: 1 The lady hit the man with an umbrella. Yi Li (Fudan University) Discrete Mathematics(II) March 27, 2017 4 / 28

Ambiguity Example Consider the following sentences: 1 The lady hit the man with an umbrella. 2 He gave her cat food. Yi Li (Fudan University) Discrete Mathematics(II) March 27, 2017 4 / 28

Ambiguity Example Consider the following sentences: 1 The lady hit the man with an umbrella. 2 He gave her cat food. 3 They are looking for teachers of French, German and Japanese. Yi Li (Fudan University) Discrete Mathematics(II) March 27, 2017 4 / 28

Ambiguity Example Consider the following proposition A 1 A 2 A 3. Yi Li (Fudan University) Discrete Mathematics(II) March 27, 2017 5 / 28

Ambiguity Example Consider the following proposition A 1 A 2 A 3. We have two possible different propositions 1 (A 1 A 2 ) A 3 Yi Li (Fudan University) Discrete Mathematics(II) March 27, 2017 5 / 28

Ambiguity Example Consider the following proposition A 1 A 2 A 3. We have two possible different propositions 1 (A 1 A 2 ) A 3 2 A 1 (A 2 A 3 ) Yi Li (Fudan University) Discrete Mathematics(II) March 27, 2017 5 / 28

Ambiguity Example Consider the following proposition A 1 A 2 A 3. We have two possible different propositions 1 (A 1 A 2 ) A 3 2 A 1 (A 2 A 3 ) Of course, they have different abbreviated truth tables. Yi Li (Fudan University) Discrete Mathematics(II) March 27, 2017 5 / 28

Count of Parentheses Theorem Every well-formed proposition has the same number of left as right parentheses. Yi Li (Fudan University) Discrete Mathematics(II) March 27, 2017 6 / 28

Count of Parentheses Theorem Every well-formed proposition has the same number of left as right parentheses. Proof. Yi Li (Fudan University) Discrete Mathematics(II) March 27, 2017 6 / 28

Count of Parentheses Theorem Every well-formed proposition has the same number of left as right parentheses. Proof. 1 Consider the symbols without parentheses first. Yi Li (Fudan University) Discrete Mathematics(II) March 27, 2017 6 / 28

Count of Parentheses Theorem Every well-formed proposition has the same number of left as right parentheses. Proof. 1 Consider the symbols without parentheses first. 2 And then prove it by induction with more complicated propositions according to the Definition. Yi Li (Fudan University) Discrete Mathematics(II) March 27, 2017 6 / 28

Prefix Theorem Any proper initial segement of a well-defined proposition contains an excess of left parenthesiss. Thus no proper initial segement of a well defined propositon can itself be a well defined propositions. Proof. Prove it by induction from simple to complicated propositions. Yi Li (Fudan University) Discrete Mathematics(II) March 27, 2017 7 / 28

Formation Tree Example The formation tree of (A B), ((A B) C). Yi Li (Fudan University) Discrete Mathematics(II) March 27, 2017 8 / 28

Formation Tree Example The formation tree of (A B), ((A B) C). How? Yi Li (Fudan University) Discrete Mathematics(II) March 27, 2017 8 / 28

Formation Tree Example The formation tree of (A B), ((A B) C). How? Form a tree bottom-up while constructing the proposition according to the Definition. Yi Li (Fudan University) Discrete Mathematics(II) March 27, 2017 8 / 28

Formation Tree Definition (Top-down) A formation tree is a finite tree T of binary sequences whose nodes are all labeled with propositions. The labeling satisfies the following conditions: Yi Li (Fudan University) Discrete Mathematics(II) March 27, 2017 9 / 28

Formation Tree Definition (Top-down) A formation tree is a finite tree T of binary sequences whose nodes are all labeled with propositions. The labeling satisfies the following conditions: 1 The leaves are labeled with propositional letters. Yi Li (Fudan University) Discrete Mathematics(II) March 27, 2017 9 / 28

Formation Tree Definition (Top-down) A formation tree is a finite tree T of binary sequences whose nodes are all labeled with propositions. The labeling satisfies the following conditions: 1 The leaves are labeled with propositional letters. 2 if a node σ is labeled with a proposition of the form (α β), (α β), (α β) or (α β), its immediate successors, σˆ0 and σˆ1, are labeled with α and β (in that order). Yi Li (Fudan University) Discrete Mathematics(II) March 27, 2017 9 / 28

Formation Tree Definition (Top-down) A formation tree is a finite tree T of binary sequences whose nodes are all labeled with propositions. The labeling satisfies the following conditions: 1 The leaves are labeled with propositional letters. 2 if a node σ is labeled with a proposition of the form (α β), (α β), (α β) or (α β), its immediate successors, σˆ0 and σˆ1, are labeled with α and β (in that order). 3 if a node σ is labeled with a proposition of the form ( α), its unique immediate successor, σˆ0, is labeled with α. Yi Li (Fudan University) Discrete Mathematics(II) March 27, 2017 9 / 28

Formation Tree Definition Yi Li (Fudan University) Discrete Mathematics(II) March 27, 2017 10 / 28

Formation Tree Definition 1 The depth of a proposition is the depth of associated formation tree. Yi Li (Fudan University) Discrete Mathematics(II) March 27, 2017 10 / 28

Formation Tree Definition 1 The depth of a proposition is the depth of associated formation tree. 2 The support of a proposition is the set of propositional letters that occur as labels of the leaves of the associated formation tree. Yi Li (Fudan University) Discrete Mathematics(II) March 27, 2017 10 / 28

Formation Tree Theorem Each well-defined proposition has a unique formation tree associated with it. Yi Li (Fudan University) Discrete Mathematics(II) March 27, 2017 11 / 28

Formation Tree Theorem Each well-defined proposition has a unique formation tree associated with it. Proof. Yi Li (Fudan University) Discrete Mathematics(II) March 27, 2017 11 / 28

Formation Tree Theorem Each well-defined proposition has a unique formation tree associated with it. Proof. 1 Existence of the formation tree by induction on depth. Yi Li (Fudan University) Discrete Mathematics(II) March 27, 2017 11 / 28

Formation Tree Theorem Each well-defined proposition has a unique formation tree associated with it. Proof. 1 Existence of the formation tree by induction on depth. 2 Uniqueness of the formation tree by induction on depth. Yi Li (Fudan University) Discrete Mathematics(II) March 27, 2017 11 / 28

Truth Assignment How we discuss the truth of propositional letters? Yi Li (Fudan University) Discrete Mathematics(II) March 27, 2017 12 / 28

Truth Assignment How we discuss the truth of propositional letters? Definition (Assignment) A truth assignment A is a function that assigns to each propositional letter A a unique truth value A(A) {T, F }. Yi Li (Fudan University) Discrete Mathematics(II) March 27, 2017 12 / 28

Truth Valuation How we discuss the truth of propositions? Yi Li (Fudan University) Discrete Mathematics(II) March 27, 2017 13 / 28

Truth Valuation How we discuss the truth of propositions? Example Truth assignment of α and β and valuation of (α β). α β (α β) T T T T F T F T T F F F Yi Li (Fudan University) Discrete Mathematics(II) March 27, 2017 13 / 28

Assignment and Valuation Definition (Valuation) A truth valuation V is a function that assigns to each proposition α a unique truth value V(α) so that its value on a compund proposition is determined in accordance with the appropriate truth tables. Yi Li (Fudan University) Discrete Mathematics(II) March 27, 2017 14 / 28

Assignment and Valuation Definition (Valuation) A truth valuation V is a function that assigns to each proposition α a unique truth value V(α) so that its value on a compund proposition is determined in accordance with the appropriate truth tables. Specially, V(α) determines one possible truth assignment if α is a propositional letter. Yi Li (Fudan University) Discrete Mathematics(II) March 27, 2017 14 / 28

Assignment and Valuation Theorem Given a truth assignment A there is a unique truth valuation V such that V(α) = A(α) for every propositonal letter α. Yi Li (Fudan University) Discrete Mathematics(II) March 27, 2017 15 / 28

Assignment and Valuation Theorem Given a truth assignment A there is a unique truth valuation V such that V(α) = A(α) for every propositonal letter α. Proof. The proof can be divided into two step. Yi Li (Fudan University) Discrete Mathematics(II) March 27, 2017 15 / 28

Assignment and Valuation Theorem Given a truth assignment A there is a unique truth valuation V such that V(α) = A(α) for every propositonal 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. Yi Li (Fudan University) Discrete Mathematics(II) March 27, 2017 15 / 28

Assignment and Valuation Theorem Given a truth assignment A there is a unique truth valuation V such that V(α) = A(α) for every propositonal 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. Yi Li (Fudan University) Discrete Mathematics(II) March 27, 2017 15 / 28

Assignment and Valuation Corollary If V 1 and V 2 are two valuations that agree on the support of α, the finite set of propostional letters used in the construction of the proposition of the proposition α, then V 1 (α) = V 2 (α). Yi Li (Fudan University) Discrete Mathematics(II) March 27, 2017 16 / 28

Tautology Definition A proposition σ of propostional logic is said to be valid if for any valuation V, V(σ) = T. Such a proposition is also called a tautology. Yi Li (Fudan University) Discrete Mathematics(II) March 27, 2017 17 / 28

Tautology Example α α is a tautology. Yi Li (Fudan University) Discrete Mathematics(II) March 27, 2017 18 / 28

Tautology Example α α is a tautology. Solution: α α α α T F T F T T Yi Li (Fudan University) Discrete Mathematics(II) March 27, 2017 18 / 28

Logical Equivenlence Definition Two proposition α and β such that, for every valuation V, V(α) = V(β) are called logically equivalent. We denote this by α β. Yi Li (Fudan University) Discrete Mathematics(II) March 27, 2017 19 / 28

Logical Equivenlence(Cont.) Example α β α β. Yi Li (Fudan University) Discrete Mathematics(II) March 27, 2017 20 / 28

Logical Equivenlence(Cont.) Example α β α β. Proof. Prove by truth table. α β α β T T T T F F F T T F F T α β α α β T T F T T F F F F T T T F F T T Yi Li (Fudan University) Discrete Mathematics(II) March 27, 2017 20 / 28

Consequence Definition Let Σ be a (possibly infinite) set of propositions. We say that σ is a consequence of Σ (and write as Σ = σ) if, for any valuation V, (V(τ) = T for all τ Σ) V(σ) = T. Yi Li (Fudan University) Discrete Mathematics(II) March 27, 2017 21 / 28

Consequence Example Yi Li (Fudan University) Discrete Mathematics(II) March 27, 2017 22 / 28

Consequence Example 1 Let Σ = {A, A B}, we have Σ = B. Yi Li (Fudan University) Discrete Mathematics(II) March 27, 2017 22 / 28

Consequence Example 1 Let Σ = {A, A B}, we have Σ = B. 2 Let Σ = {A, A B, C}, we have Σ = B. Yi Li (Fudan University) Discrete Mathematics(II) March 27, 2017 22 / 28

Consequence Example 1 Let Σ = {A, A B}, we have Σ = B. 2 Let Σ = {A, A B, C}, we have Σ = B. 3 Let Σ = { A B}, we have Σ = B. Yi Li (Fudan University) Discrete Mathematics(II) March 27, 2017 22 / 28

Model Definition We say that a valuation V is a model of Σ if V(σ) = T for every σ Σ. We denote by M(Σ) the set of all models of Σ. Yi Li (Fudan University) Discrete Mathematics(II) March 27, 2017 23 / 28

Model Example Let Σ = {A, A B}, we have models: Yi Li (Fudan University) Discrete Mathematics(II) March 27, 2017 24 / 28

Model Example Let Σ = {A, A B}, we have models: 1 Let A(A) = T, A(B) = T Yi Li (Fudan University) Discrete Mathematics(II) March 27, 2017 24 / 28

Model Example Let Σ = {A, A B}, we have models: 1 Let A(A) = T, A(B) = T 2 Let A(A) = T, A(B) = T, A(C) = T. Yi Li (Fudan University) Discrete Mathematics(II) March 27, 2017 24 / 28

Model Example Let Σ = {A, A B}, we have models: 1 Let A(A) = T, A(B) = T 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,.... Yi Li (Fudan University) Discrete Mathematics(II) March 27, 2017 24 / 28

Model Definition We say that propositions Σ is satisfiable if it has some model. Otherwise it is called unsatisfiable. To a proposition, it is called invalid. Yi Li (Fudan University) Discrete Mathematics(II) March 27, 2017 25 / 28

Properties Proposition Let Σ, Σ 1, Σ 2 be sets of propositions. Let Cn(Σ) denote the set of consequence of Σ and Taut the set of tautologies. Yi Li (Fudan University) Discrete Mathematics(II) March 27, 2017 26 / 28

Properties Proposition Let Σ, Σ 1, Σ 2 be sets of propositions. Let Cn(Σ) denote the set of consequence of Σ and Taut the set of tautologies. 1 Σ 1 Σ 2 Cn(Σ 1 ) Cn(Σ 2 ). Yi Li (Fudan University) Discrete Mathematics(II) March 27, 2017 26 / 28

Properties Proposition Let Σ, Σ 1, Σ 2 be sets of propositions. Let Cn(Σ) denote the set of consequence of Σ and Taut the set of tautologies. 1 Σ 1 Σ 2 Cn(Σ 1 ) Cn(Σ 2 ). 2 Σ Cn(Σ). Yi Li (Fudan University) Discrete Mathematics(II) March 27, 2017 26 / 28

Properties Proposition Let Σ, Σ 1, Σ 2 be sets of propositions. Let Cn(Σ) denote the set of consequence of Σ and Taut the set of tautologies. 1 Σ 1 Σ 2 Cn(Σ 1 ) Cn(Σ 2 ). 2 Σ Cn(Σ). 3 Taut Cn(Σ) = Cn(Cn(Σ)). Yi Li (Fudan University) Discrete Mathematics(II) March 27, 2017 26 / 28

Properties Proposition Let Σ, Σ 1, Σ 2 be sets of propositions. Let Cn(Σ) denote the set of consequence of Σ and Taut the set of tautologies. 1 Σ 1 Σ 2 Cn(Σ 1 ) Cn(Σ 2 ). 2 Σ Cn(Σ). 3 Taut Cn(Σ) = Cn(Cn(Σ)). 4 Σ 1 Σ 2 M(Σ 2 ) M(Σ 1 ). Yi Li (Fudan University) Discrete Mathematics(II) March 27, 2017 26 / 28

Properties Proposition Let Σ, Σ 1, Σ 2 be sets of propositions. Let Cn(Σ) denote the set of consequence of Σ and Taut the set of tautologies. 1 Σ 1 Σ 2 Cn(Σ 1 ) Cn(Σ 2 ). 2 Σ Cn(Σ). 3 Taut Cn(Σ) = Cn(Cn(Σ)). 4 Σ 1 Σ 2 M(Σ 2 ) M(Σ 1 ). 5 Cn(Σ) = {σ V(σ) = T for all V M(Σ)}. Yi Li (Fudan University) Discrete Mathematics(II) March 27, 2017 26 / 28

Properties Proposition Let Σ, Σ 1, Σ 2 be sets of propositions. Let Cn(Σ) denote the set of consequence of Σ and Taut the set of tautologies. 1 Σ 1 Σ 2 Cn(Σ 1 ) Cn(Σ 2 ). 2 Σ Cn(Σ). 3 Taut Cn(Σ) = Cn(Cn(Σ)). 4 Σ 1 Σ 2 M(Σ 2 ) M(Σ 1 ). 5 Cn(Σ) = {σ V(σ) = T for all V M(Σ)}. 6 σ Cn({σ 1,..., σ n }) σ 1 (σ 2... (σ n σ)...) Taut. Yi Li (Fudan University) Discrete Mathematics(II) March 27, 2017 26 / 28

Deduction Theorem Theorem For any propositions φ, ψ, Σ {ψ} = φ Σ = ψ φ holds. Yi Li (Fudan University) Discrete Mathematics(II) March 27, 2017 27 / 28

Deduction Theorem Theorem 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, V 1 (ψ) = T and V 2 (ψ) = F. Then we investigate whether V satisfies ψ φ. Conversely, V which makes ψ false are discarded. Because they are not taken into consideration to satisfy Σ {ψ}. Yi Li (Fudan University) Discrete Mathematics(II) March 27, 2017 27 / 28

Next Class Tableau Proof Yi Li (Fudan University) Discrete Mathematics(II) March 27, 2017 28 / 28