Unit 4 Propositional logic Discrete Mathematics
- 2. Propositional Logic Proposition : Proposition is a statement which is either true or false but not both. It is a declarative statement. It is usually denoted by lower case letters p, q, r, s, t etc.They are called Boolean variable or logic variable. For example : "Man is Mortal", it returns truth value “TRUE” "12 + 9 = 3 – 2", it returns truth value “FALSE
- 3. Compound proposition A compound proposition is formed by composition of two or more propositions called components or sub- propositions. For example : 1. Rishabh is intelligent and he studies hard. 2. Sky is blue and clouds are white. Here first statement contain two propositions ‘‘Rishabh is intelligent’’ and ‘‘he studies hard’’ whereas second statement contain propositions ‘‘sky is blue’’ and ‘‘clouds are white’’. As both statements are formed using two propositions. So they are compound propositions.
- 4. Compound Propositions The words or phrases used to form compound proposition are called logical connectives. The following are the logical connectives: Negation (not) p Conjunction (and) p q Disjunction (or) p q Conditional p q Biconditional p q
- 5. 5 Truth tables Since we need to know the truth value of a proposition in all possible scenarios, we consider all the possible combinations of the propositions which are joined together by Logical Connectives to form the given compound proposition. This compilation of all possible scenarios in a tabular format is called a truth table. A truth table is a table that shows the truth value of a compound proposition for all possible cases.
- 6. Propositions and their negations If p is a proposition then negation of p is a proposition which is true when p is false and false when p is true. It is denoted by ~ p or ¬ p . Examples p: It is raining. ∼ p: It is not raining. p: 3+2=5. ∼ p: 3+2≠5. Negation p ∼p T F F T
- 7. 7 Conjunctions Consider two propositions, p and q Conjunction (“and”): p Λ q It is a bright and windy day. The day has to be both bright and windy. Example : p : Ram is healthy. q : He has blue eyes. p Λ q : Ram is healthy and he has blue eyes. Conjunction (“and”) p q p Λ q F F F F T F T F F T T T
- 8. 8 Disjunctions Consider two propositions, p and q Disjunction (“or”): p V q To ride the bus you must have a ticket or hold a pass. •One of the two conditions (“have a ticket” or “hold a pass”) suffices. (Though both could be true.) Example : p : Ram will go to Delhi. q : Ram will go to Calcutta. p V q : Ram will go to Delhi or Calcutta. Disjunction (“or”) p q p V q F F F F T T T F T T T T
- 9. Logical equivalence Two statements are logically equivalent if and only if they have identical truth tables. The simplest example is ∼(∼p) ≡ p. p ∼p ∼(p) F T F T F T
- 10. Algebra of Propositions Proposition satisfies various laws which are useful in simplifying complex expressions. These laws are listed as : 1. Identity laws : a. p T ≡ p ˄ b. p F ≡ p ˅ 2. Domination laws: c. p F ≡ F ˄ d. p T ≡ T ˅ 3. Idempotent laws : e. p p ˅ ≡ p f. p p ˄ ≡ p
- 11. Algebra of Propositions 4. Commutative laws : a. p q ˅ ≡ q p ˅ b. p q ˄ ≡ q p ˄ 5. Associative laws : c. (p q) r ˅ ˅ ≡ p (q r) ˅ ˅ d. (p q) r ˄ ˄ ≡ p (q r) ˄ ˄ 6. Distributive laws : e. p (q r) ≡ (p q) (p r) ˅ ˄ ˅ ˄ ˅ f. p (q r) ≡ (p q) (p r) ˄ ˅ ˄ ˅ ˄ 7. Absorption laws : g. p (p q) ≡ p ˅ ˄ h. p (p q) ≡ p ˄ ˅
- 12. Algebra of Propositions 8. Complement laws : a. p ~P ≡ T ˅ b. p ~p ≡ F ˄ c. ~T ≡ F d. ~F ≡ T 8. Involution law : e. ~(~p) ≡ p 9. De Morgan's laws : f. ~ (p q) ≡ ~p ~q ˅ ˄ g. ~ (p q) ≡ ~p ~q ˄ ˅
- 13. 13 Conditional Proposition If p and q are proposition, the compound proposition “if p then q” denoted by p → q is called conditional Proposition. The proposition p is called antecedent or hypothesis, and the proposition q is called the consequent or conclusion. The only circumstances under which p → q is false when p is true and q is false. Example: If tomorrow is Sunday then today is Saturday p:Tomorrow is Sunday q:Today is Saturday
- 14. 14 Truth table for Conditional Proposition The connectives if _____ then ca be read as: p is sufficient for q. p only if q p is necessary for q q if p p is consequence of q Conditional p q p → q T F F T T T F F T F T T
- 15. An equivalent for Conditional Is there an expression that is equivalent to p → q? Implication p q p → q F F T F T T T F F T T T p q ∼p ∼p V q F F T T F T T T T F F F T T F T Consider the proposition ∼p V q
- 16. 16 Variations on a proposition Given a proposition p → q, there are other propositions that can be stated. ◦ Example: If it rains then the crop will grow. ◦ p: It rains ◦ q: The crop will grow Converse: q → p ◦ Example: If the crop grow, then there has been rain Contrapositive: ¬q → ¬p ◦ Example: If the crops do not grow, then there has been no rain. Inverse: ¬p → ¬q ◦ Example: If it does not rain then the crop will not grow .
- 17. 17 Variations on a proposition A conditional proposition and its converse or inverse are not logically equivalent. But a conditional proposition and its contrapositive are logically equivalent. ~ q → ~ p ≡ p → q Conditional p q ~p ~q ~q → ~p p → q T T F F T T T F F T F F F T T F T T F F T T T T
- 18. 18 Biconditional Proposition If p and q are proposition , then the compound proposition p if and only if q ,denoted by p q, is called biconditional Propositional. The biconditional proposition p q can be stated as “ p is necessary and sufficient condition for q”, “p iff q”. Example: 1. He swims if and only if the water is warm. 2. Sales of house falls if and only if interest rate rises.
- 19. 19 Biconditional Proposition The truth table of p q is as follows: It may be noted p q is true when both p and q are true or both p and q are false Biconditional p q p q T T T T F F F T F F F T
- 20. 20 Other derived connectives NAND: It means negation of conjunction of two statements . If p and q are two proposition , the NAND of p and q is false when both p and q are true otherwise It is denoted by p ↑ q p ↑ q ≡∼(p q) ˄ NAND p q p ↑ q T T F T F T F T T F F T
- 21. 21 Other derived connectives NOR: It means negation of disjunction of two statements. If p and q are two proposition , the NAND of p and q is true when both p and q are false otherwise It is denoted by p ↓ q p ↓ q ≡∼(p q) ˅ NAND p q p ↑ q T T F T F T F T T F F T
- 22. 22 Other derived connectives XOR (Exclusive OR): If p and q are two proposition , the XOR of p and q, denoted by p ⊕ q is true when exactly one of p and q is true. It is denoted by p ⊕ q XOR p q p ⊕ q T T F T F T F T T F F F
- 23. Well formed formula Well formed formula is an expression which consist of variables, parenthesis ad connective symbols, braces and square brackets to avoid the ambiguity. Logical statements are also represented in well defined form using parenthesis according to priority of operations.
- 24. Well formed formula To generate well formed formula recursively, following rules are used : i. An atomic statement P is a well formed formula. ii. If P is well formed formula then ~ P is also a well formed. iii. If P and Q are well formed formulae then (P˅Q), (P˄Q), (P→Q) and (P↔Q) are also well formed formulae. iv. A statement consists of variables, parenthesis and connectives is recursively well formed formula iff it can be obtained by applying the above three rules. For example : 1. (P → (P ˅ Q)) is a well formed formula. 2. (P ˄ Q) → (˅ R) is not a well formed formula.
- 25. Principle of Duality Two formulas A1 and A2 are said to be duals of each other if either one can be obtained from the other by replacing ˄ (AND) by ˅ (OR) and vice versa. Also if formula contains T(True) and F(False), then we replace T by F and vice versa. Example The dual of (A − ˄ B ) ˅ C is (A ˅ B) ˄ C
- 26. Normal Forms We can convert any proposition in two normal forms − Disjunctive Normal Form (DNF) Conjunctive Normal Form (CNF) Principal Disjunctive Normal Form (PDNF) Principal Conjunctive Normal Form (PCNF)
- 27. Disjunctive Normal Form A compound statement is in conjunctive normal form if it is obtained by operating OR among variables (negation of variables included) connected with ANDs. In terms of set operations, it is a compound statement obtained by Union among variables connected with Intersections. Examples p (q r) and p ( ∨ ∧ ∨ ~ q r) are in DNF, ∧ but p (q r) is not a DNF. ∧ ∨
- 28. Method to construct DNF Construct a truth table for the proposition. Use the rows of the truth table where the proposition is True to construct minterms ◦ If the variable is true, use the propositional variable in the minterm ◦ If a variable is false, use the negation of the variable in the minterm Connect the minterms with ’s.
- 29. 29 Disjunctive normal form (DNF) p q p q → T T T T F F F T T F F T pq pq p q pq (pq) ( pq) ( p q) OR
- 30. Conjunctive Normal Form A compound statement is in conjunctive normal form if it is obtained by operating AND among variables (negation of variables included) connected with ORs. In terms of set operations, it is a compound statement obtained by Intersection among variables connected with Unions. Examples (A B) (A C) (B C D) ∨ ∧ ∨ ∧ ∨ ∨ (P Q) (Q R) ∪ ∩ ∪
- 31. Tautology Tautology is defined as a compound proposition that is always true for all possible truth values of its propositional variables and it contains T in last column of its truth table. Propositions like, i. The doctor is either male or female. ii. Either it is raining or not. are always true and are tautologies.
- 32. Tautology For Example: p ∼p ˅ Since the truth value is TRUE for all possible values, so the proposition is a tautology. p ∼p p p ˅∼ T F T F T T
- 33. Tautology Check whether the given propositions are contradiction. 1. ∼ (p q) q ˅ ˄ 2. p → ( p q) ˅
- 34. Contradiction : Contradiction is defined as a compound proposition that is always false for all possible truth values of its propositional variables and it contains F in last column of its truth table. Propositions like, i. x is even and x is odd number. ii. Tom is good boy and Tom is bad boy. are always false and are contradiction.
- 35. Contradiction : For Example: p (q ∼p) ˄ ˅ Since the truth value is FALSE for all possible values, so the proposition is a Contradiction. p Q ∼p q p ˅∼ p˄(q p) ˅∼ T T F F F T F F F F F T T T F F F T F F
- 36. Contradiction : Check whether the given propositions are contradiction. 1. ∼ (p q) q ˅ ˄ 2. p → ( p q) ˅
- 37. Contingency A Contingency is a formula which has both some true and some false values for every value of its propositional variables. A proposition that is neither a tautology nor contradiction is called a contingency.
- 38. Satisfiability : A compound statement formula A (P1 , P2 , ... Pn ) is said to be satisfiable, if it has the truth value T for at least one combination of truth value of P1 , P2 ,.... Pn .
- 39. Satisfiability : Example Prove (A B) (¬A) is satisfiable − ∨ ∧ The truth table is as follows − As we can see at least one value of (A B) (¬A) is “True” , it is a ∨ ∧ satisfiablity.
- 40. 40 What is an argument? A sequence of statements the ends with a conclusion. (Not the common language usage of a debate or dispute.) Structure of an argument Statement 1 (p1) Statement 2 (p2) ... Statement n (pn) Statement n+1 (conclusion) premises or antecedents
- 41. 41 Example Premises “If you have a current password, you can log onto the computer network.” “You have a current password.” Conclusion “Therefore, you can log onto the computer network.”
- 42. 42 Representing an argument An argument is sometimes written as follows: premises conclusion
- 43. 43 Valid arguments An argument is valid if and only if it is impossible for all the premises to be true and the conclusion to be false. How do we show that an argument is valid? We can use a truth table, or We can show that (p1 Λ p2 Λ ... Λ pn → pn+1) is a tautology using some rules of inference.
- 44. 44 Rules of Inference The rules of Inference are criteria for determining the validity of an argument. Any conclusion which is arrived by following the rules is called a valid conclusion, and the argument is called valid argument.
- 45. 45 Why use rules of inference? Constructing a truth table is time consuming! If we have n propositions, what is the size of the truth table? 2n , which means that the table doubles in size with every proposition. Two propositions are involved in an implication, therefore the truth table has 22 = 4 rows. Implication p q p → q F F T F T T T F F T T T
Editor's Notes
- #46: Many valid inference rules were first described by Aristotle. He called these patterns of argument “syllogisms.”