![Propositional Logic - II
[Universal and Existential Quantifiers]
Maham Noor](https://image.slidesharecdn.com/propositionallogic-250120183352-c3b41c24/85/Propositional-Logic-pptx-discrete-mathmatics-1-320.jpg)
Propositional Logic.pptx discrete mathmatics
- 1. Propositional Logic - II [Universal and Existential Quantifiers] Maham Noor
- 2. Propositional Logic Vs Predicate Logic •Propositional Logic deals with statements that are true or false but does not involve variables or quantifiers. For example, "It is raining" (true/false). •Predicate Logic introduces variables, predicates, and quantifiers to express more complex statements about objects and their properties, like "Some stones are precious."
- 3. English-to-Symbolic Form • Some stones are precious. • All grapes are red. • There is an honest politician • Some celebrities are invited. • Few students are present.
- 4. English-to-Symbolic Form Some stones are precious. This is a logical assertation about a set of objects (stones) and a property (being precious) In predicate logic we represent such statements using quantifiers and predicates x: set of stones (universe of discourse) x: represents the set of stones. This is the domain of the discourse (all objects we are talking about) Precious (x) : A predicate that means x is precious Representation in Predicate Logic: Some stones are precious is represented as : existential quantifier which means “there exists” x refers to an element in the set of stones Precious (x): asserts that x has the property of being precious Together, x Precious(x) means "There exists at least one stone x such that x is precious." ∃
- 5. English-to-Symbolic Form • All grapes are red. x: set of grapes (universe of discourse) Red (x) : x is a red grape all grapes are red:
- 6. English-to-Symbolic Form • All students are hungry. x: set of students (universe of discourse) Hungry (x) : x is a hungry student all students are hungry:
- 7. Propositional Function - Predicate
- 9. Quantifiers
- 11. Example Let P(x) be “x must take Discrete Mathematics course”, And let Q(x) be “x is a Computer Science student”. The Universe of discourse is all UCP students. • Express the statement mathematically: “Every computer science student must take discrete mathematics course”
- 12. Example • Let be “x must take Discrete Mathematics course”, And let be “x is a Computer Science student”. The Universe of discourse is all UCP students (). • Express the statement: “Every computer science student must take discrete mathematics course”
- 13. Example • Let P(x) be “x must take Discrete Mathematics course”, And let Q(x) be “x is a Computer Science student”. The Universe of discourse is all UCP students. • Express the statement: “Every computer science student must take discrete mathematics course” • Express the statement: “Everybody must take a discrete mathematics course or be a computer science student”
- 14. Example • Let P(x) be “x must take Discrete Mathematics course”, And let Q(x) be “x is a Computer Science student”. The Universe of discourse is all UCP students. • Express the statement: “Every computer science student must take discrete mathematics course” • Express the statement: “Everybody must take a discrete mathematics course or be a computer science student” • ∀x(P(x) Q(x)) ∨
- 15. Example • Let P(x) be “x must take Discrete Mathematics course”, And let Q(x) be “x is a Computer Science student”. The Universe of discourse is all UCP students. • Express the statement: “Every computer science student must take discrete mathematics course” • Express the statement: “Everybody must take a discrete mathematics course or be a computer science student”. • ∀x(P(x) Q(x)) ∨ • Are these statements True or False?
- 17. Truth Values of Quantifiers
- 18. Examples – Express in Mathematical Notation • Product of two negative integers is positive.
- 19. Examples – Express in Mathematical Notation • Product of two negative integers is positive.
- 20. Examples – Express in Mathematical Notation • Product of two negative integers is positive. • Average of two positive integers in positive.
- 21. Examples – Express in Mathematical Notation • Product of two negative integers is positive. • Average of two positive integers is positive.
- 22. Examples – Express in Mathematical Notation • Product of two negative integers is positive. • Average of two positive integers in positive. • The difference of two negative integers is not necessarily negative.
- 23. Examples – Express in Mathematical Notation • Product of two negative integers is positive. • Average of two positive integers in positive. • The difference of two negative integers is not necessarily negative. Or we may write
- 24. Examples – Express in Mathematical Notation • Product of two negative integers is positive. • Average of two positive integers in positive. • The difference of two negative integers is not necessarily negative. • Absolute value of sum of two integers does not exceed the sum of the absolute values of these integers.
- 26. Quantifiers – Negation Use De Morgan’s Law
- 27. Quantifiers – Negation Use De Morgan’s Law
- 28. Quantifiers – Negation – Truth Values
- 29. Example – 1
- 30. Example – 1 • Negation All politicians are dishonest Writing in mathematical notation: : set of all politicians is honest There is an honest politician: Negation: All politicians are dishonest:
- 31. Example – 1 • Negation There is an American who does not eat cheeseburger : set of Americans : eats Cheeseburger Negation:
- 32. Example – 2 • 2nd statement
- 33. Mixing Quantifiers – Nested Quantifiers
- 34. Mixing Quantifiers Everybody loves everyone Everybody loves somebody Somebody loves somebody
- 35. Mixing Quantifiers is a person is a country National: is national of : National Everyone belongs to some country : National There is a country, everybody belongs to this country.
- 36. Nested Quantifiers – Truth Values • Negation
- 37. Example – 3 Implication is equivalent to NOT P(x) or Q(x) Use De Morgan’s Law
- 38. Example – 4
- 39. Example – 4 : Students in the class has visited Mexico ================== == has visited Mexico has visited Canada : Students is student in this class has visited Mexico ==================== has visited Mexico has visited Canada
- 40. Example – 5
- 41. Example – 5 • 1st statement mail message is compressed -----------------------------
- 42. Example – 5 • 1st statement mail message is compressed -------------------- • 2nd statement user, network link is an active user is available to user
- 43. Exercises – Lewis Carroll (Alice in Wonderland)
- 44. Exercises – Lewis Carroll (Alice in Wonderland) creatures is a lion is fierce drinks coffee
- 45. A)
- 46. • No doctors are enthusiastic; • You are enthusiastic. • Therefore, you are not a doctor a person is a Doctor is enthusiastic
- 47. a) For all real numbers, there is a real number, which is greater than these numbers. (not beautiful statement) There are infinite real numbers. b) Product of two non-negative real numbers is non-negative. c) For any two real numbers, their product is a real number. (more beautiful way) Real number are closed with respect to multiplication. (closure property)
- 48. a) b) . c) .