Propositional Calculus-Discrete Mathematics
- 2. Propositional Logic Logic: A study of principles or methods used to establish/validate an argument. Proposition: An affirmation or a declarative sentence which is either true or false is called a proposition or statement . Ex: 1) 2+3=5 2) Today is sunday
- 3. Truth Value A proposition is assigned with a truth value which declares its validity. If a proposition is true then it is assigned a truth value true. If a proposition is false then it is assigned a truth value false. Statement/Proposition Truth Value Ex: TCS is a Multinational Company True 4+6=8 False
- 4. Which is not a Proposition? ● A non declarative statement is not a proposition Ex: Close the door X is even #is not declarative when we do not know What is x? She is a generous lady. #is not declarative when we do not know Who is she? ● A question is not a proposition Ex: Is it a holiday tomorrow? ● An exclamation is not a proposition Ex: Oh, That is excellent!
- 5. Think & Tell Which of the following is a statement. x+1 is odd Computer scientists require logical reasoning x+y=4 Does the teacher allow me into class? -2 < 0 My God! Those are some wonderfully great decorations.
- 6. Think & Tell Which of the following is a statement. x+1 is odd NO Computer scientists require logical reasoning YES x+y=4 NO Does the teacher allow me into class? NO -2 < 0 YES My God! Those are some wonderfully great decorations. NO
- 7. Propositional Variables In analogy with numeric variables of arithmetic logic, propositional variables are used to represent propositions/statements in Propositional logic. The lower alphabets such as p,q,r,s,t,u are used as propositional variables Ex: p: Today is sunday q: 2+3=5
- 8. Truth table The truth table of a statement displays or depicts the truth value of a statement based on the truth values of its constituent statements. In a Truth table a True value is represented by T/1 and a False value is represented by a F/0. For given n variables a truth table consists of Truth table entries.
- 9. Compound Statements Compound Statements(aka Complex Propositions) can be obtained by either transforming or combining the simple propositions/statements by using logical connectives. Transforming a Proposition Apply Negation(¬) to a statement to change its meaning Ex: A given proposition p: Today is a holiday can be transformed into another statement by applying negation ¬p: Today is not a holiday Truth Table: p ¬p 1 0 0 1
- 10. Compound Statements Combining simpler propositions Two or more simple propositions can be combined to form a new proposition using the following logical connectives. Logical Connective Symbol Conjunction ∧ Disjunction ∨ Conditional → Biconditional ↔
- 11. Logical connectives - Conjunction If p,q are two statements then the conjunction of p,q is given by “p and q” which is written as p ∧ q. The truth table of a Conjunction is given below. Truth Table: p q p ∧ q 0 0 0 0 1 0 1 0 0 1 1 1 From the truth table it is evident that p ∧ q is True only when both the constituent statements p,q are True. Ex: 2+3>4 and 4+5<10 p q
- 12. Logical connectives - Disjunction If p,q are two statements then the disjunction of p,q is given by “p or q” which is written as p ∨ q. The truth table of a Disjunction is given below. Truth Table: p q p ∨ q 0 0 0 0 1 1 1 0 1 1 1 1 From the truth table it is evident that p ∨ q is True when at least one of the constituent statements p,q are True. Ex: It may rain today or it may be sunny today p q
- 13. Logical connectives - Conditional(implication) If p,q are two statements then the conditional implication of p,q is given by “p implies q” which is written as p → q. The truth table of implication is given below. Other names for conditional implication are ● If p then q Truth Table: ● p is sufficient for q ● p is sufficient condition for q ● q is necessary for p ● q is necessary condition for p ● p only if q p q p →q 0 0 1 0 1 1 1 0 0 1 1 1 From the truth table it is evident that p → q is False only when p is True and q is False which means that we can not accept a False statement leading to a True statement.
- 14. Example for Conditional Implication p: Farooq got admission in Engineering q: Farooq has passed in his intermediate p → q : if Farooq got admission in Engineering then Farooq has passed his intermediate. This becomes false only when Farooq gets admission in engineering when Farooq has not passed his intermediate. So we say q is necessary for p to become true(or valid).
- 15. Inverse,Converse and Contrapositive ● The inverse of p → q is ¬p → ¬q. ● The converse of p → q is q → p. ● The contrapositive of p → q is ¬q → ¬p.
- 16. Logical connectives - Biconditional(Bi-implication) If p,q are two statements then the biconditional implication of p,q is given by “p bi-implies q” which is written as p ↔ q. The truth table of implication is given below. Other names for conditional implication are ● P if and only if (iff) q ● p is necessary and sufficient condition for q Truth Table: p q p↔q 0 0 1 0 1 0 1 0 0 1 1 1 From the truth table it is evident that p ↔ q is True only when both p and q have the same Truth values(True/ False)
- 17. Example for Bi-conditional Implication p: This year has 366 days q: This year is a leap year p ↔ q : This year has 366 days iff (if and only if) This year is a leap year. If it is not a leap year it can not have 366 days. So if only one statement is true here that leads to a false proposition. Therefor we say q is necessary and sufficient for p to become true(or valid).
- 18. Example Problems •(Grimaldi) Exercise 1.2.2: Translating English statements into logic. Express each statement in logic using the variables: –p: It is windy q: It is cold r: It is raining •It is windy and cold. p ∧ q •It is windy but not cold. p ∧ ¬q •It is not true that it is windy or cold. ¬(p ∨ q) •It is raining and it is windy or cold. r ∧ (p ∨ q) •It is raining and windy or it is cold. (r ∧ p) ∨ q •It is raining and windy but it is not cold. r ∧ p ∧ ¬q
- 19. Example Problems 1. To take discrete mathematics, you must have taken calculus or a course in computer science. 2. When you buy a new car from Acme Motor Company, you get $2000 back in cash or a 2% car loan. 3. School is closed if more than 2 feet of snow falls and if the wind chill is below -80.
- 20. To take discrete mathematics, you must have taken calculus or a course in computer science. Propositions - p: take discrete math - q: you have taken calculus - r : you have taken a course in CS Sol:
- 21. When you buy a new car from Acme Motor Company, you get $2000 back in cash or a 2% car loan. Propositions - p: you buy a car - q: you get $2000 back - r : you get 2% car loan Sol:
- 22. School is closed if more than 2 feet of snow falls and if the wind chill is below -80. Propositions - p: School is closed - q: 2 feet of snow falls - r : wind chill is below -80 Sol:
- 23. Types of Propositions Propositional qualities or types of Propositions are ● Tautologies ● Contradiction ● Contingency Tautology: A compound proposition is called a tautology if its truth value is True for all truth value assignments of its constituent statements. Ex:
- 24. Types of Propositions Contradiction: A compound proposition is called a Contradiction if its truth value is False for all truth value assignments of its constituent statements. Ex:
- 25. Types of Propositions Contingency: A compound proposition is called a contingency if its truth value is True for some truth value assignments of its constituent statements and False for others. Ex:
- 26. Properties/Nature of Propositions Compound Propositions can be ● Satisfiable ● Unsatisfiable ● Valid ● Invalid Satisfiable: If the given compound statement has at least one True result in its Truth table it is said to be Satisfiable. Unsatisfiable: If the given compound statement has no True result in its Truth table it is said to be Unsatisfiable. Valid: If the given compound statement is a tautology it is said to be valid. Invalid: If the given compound statement is not a tautology it is said to be invalid.
- 27. Properties of Compound Propositions Every Tautology is also Satisfiable. However, Satisfiability doesn't imply Tautology. Another thing to note is, if a propositional statement is Tautology, then its always valid. Thus, Tautology implies ( Satisfiability + Validity ).
- 28. Example Problems Find out whether the following compound proposition is satisfiable or not: (p∨¬q)∧(q∨¬r)∧(r∨¬p) Is the following proposition Satisfiable? Is it Valid? p → q ↔ q → r
- 29. Logical Equivalence Definition: Two statements s 1 and s 2 are logically equivalent, s 1 ≡ s 2 , if their truth tables are the same –i.e. s 1 is true (respectively, false) if and only if the statement s 2 is true (respectively, false). In otherwards, the compound statements having same truth values in all cases are said to be logically equivalent.
- 30. Proving the equivalence of statements by Truth table State whether is a tautology or not or Check the equivalence of and From the truth table it is evident that the given statements are logically equivalent.
- 31. Example Prove the following logical equivalence using Truth table a) b)
- 32. p q r 0 0 0 1 0 0 0 1 0 0 0 1 1 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 1 0 1 1 1 0 1 1 0 0 1 1 0 0 1 0 1 0 1 1 1 0 1 1 0 1 1 0 0 1 0 0 1 0 1 1 1 0 1 1 1 0 1
- 33. Logical equivalence of two English statements. Express each pair of sentences using logical expressions. Then prove whether the two expressions are logically equivalent. –If Sally did not get the job, then she was late for her interview or did not update her resume. If Sally updated her resume and did not get the job, then she was late for her interview.
- 34. Define the propositions: –j: Sally got the job. –l: Sally was late for her interview –r: Sally updated her resume. Express each pair of sentences using logical expressions. Then prove whether the two expressions are logically equivalent. Solution: ¬j → (l ∨ ¬r) (r ∧ ¬j) → l
- 35. Proving the equivalence between the two statements using truth table
- 36. Example Problem Express each pair of sentences using logical expressions. Then prove whether the two expressions are logically equivalent. If a person is 18 years old then he is eligible for voting. It is not true that a person who is 18 years old is not eligible for voting
- 38. Laws of logic For any primitive statements p,q,r and any tautology T0 and Contradiction F0. These laws hold (Note consider F as F0 and T as T0 in the table) Negation laws are also called Inverse laws
- 39. These equivalences can be proved by using the Laws of Logic. Take it as exercise for practice
- 40. Proving the logical equivalence of the statements using laws of logic Provide the steps and reasons for establishing the following equivalences
- 43. Prescribed Textbooks More exercise problems can be found in 1. Discrete and Combinatorial Mathematics by Ralph P.Grimaldi https://drive.google.com/file/d/1ttjn6ZD6FeX8X_NydAABZwawrNNnALnQ/view?usp=drive_link 2. Discrete Mathematics And Its Applications by Kenneth H Rosen https://drive.google.com/file/d/1gRHU9fr9_JT9mt_FkSGiUXXjWZuqebA7/view?usp=drive_link