Discrete Structures Predicate Logic 2

1. Discrete StructuresPredicate Logic 2 Dr. Muhammad Humayoun Assistant Professor COMSATS Institute of Computer Science, Lahore. mhumayoun@ciitlahore.edu.pk https://sites.google.com/a/ciitlahore.edu.pk/dstruct/

2. Negation of Quantifiers • ???

3. Negation of Quantifiers • ???

4. Negation of Quantifiers • ???

5. Exercise B(x): “x is a baby” ignorant(x): “x is ignorant” vain(x): “x is vain” Universe: The set of all people. • Babies are ignorant.

6. Exercise B(x): “x is a baby” ignorant(x): “x is ignorant” vain(x): “x is vain” Universe: The set of all people. • Babies are ignorant. (Ambiguous) • All/Some babies are ignorant

7. Exercise B(x): “x is a baby” ignorant(x): “x is ignorant” vain(x): “x is vain” Universe: The set of all people. • Babies are ignorant. (Ambiguous) • All babies are ignorant

8. Exercise professor(x): “x is a professor” ignorant(x): “x is ignorant” vain(x): “x is vain” Universe: The set of all people. • No professors are ignorant. • It is not the case that there exists an x such that x is a professor and x is ignorant. • It is not the case that all professors are ignorant.

9. Exercise professor(x): “x is a professor” ignorant(x): “x is ignorant” vain(x): “x is vain” Universe: The set of all people. • No professors are ignorant. • [There is no such professor who is ignorant] • [It is not the case that there is an x such that x is a professor and x is ignorant.] • It is not the case that all professors are ignorant.

10. Exercise professor(x): “x is a professor” ignorant(x): “x is ignorant” vain(x): “x is vain” Universe: The set of all people. • No professors are ignorant. • [There is no such professor who is ignorant] • [It is not the case that there is an x such that x is a professor and x is ignorant.] professors are ignorant.

11. Exercise professor(x): “x is a professor” ignorant(x): “x is ignorant” vain(x): “x is vain” Universe: The set of all people. • No professors are ignorant. • [There is no such professor who is ignorant] • [It is not the case that there is an x such that x is a professor and x is ignorant.] • All professors are not ignorant

12. Exercise professor(x): “x is a professor” ignorant(x): “x is ignorant” vain(x): “x is vain” Universe: The set of all people. • No professors are ignorant. • All (and all of them) professors are not ignorant.

13. Exercise professor(x): “x is a professor” ignorant(x): “x is ignorant” vain(x): “x is vain” Universe: The set of all people. • All ignorant people are vain. • For all people x, if x is ignorant then x is vain. • It is logically equivalent to • There is no such person x such that he is ignorant and not vain.

14. Exercise professor(x): “x is a professor” ignorant(x): “x is ignorant” vain(x): “x is vain” Universe: The set of all people. • All ignorant people are vain. • For all people x, if x is ignorant then x is vain. • It is logically equivalent to • There is no such person x such that he is ignorant and not vain.

15. Exercise professor(x): “x is a professor” ignorant(x): “x is ignorant” vain(x): “x is vain” Universe: The set of all people. • All ignorant people are vain. • For all people x, if x is ignorant then x is vain. • It is logically equivalent to • There is no such person x such that he is ignorant and not vain.

16. Exercise professor(x): “x is a professor” ignorant(x): “x is ignorant” vain(x): “x is vain” Universe: The set of all people. • All ignorant people are vain. • For all people x, if x is ignorant then x is vain. • It is logically equivalent to • There is no such person x such that he is ignorant and not vain.

17. Exercise professor(x): “x is a professor” ignorant(x): “x is ignorant” vain(x): “x is vain” Universe: The set of all people. • All ignorant people are vain. • For all people x, if x is ignorant then x is vain. • It is logically equivalent to • There is no such person x such that he is ignorant and not vain. Useful

18. Exercise professor(x): “x is a professor” ignorant(x): “x is ignorant” vain(x): “x is vain” Universe: The set of all people. • No professors are vain • It is not the case that there is an x such that x is professor and x is vain. • For all people x, if x is a professor then x not vain.

19. Exercise professor(x): “x is a professor” ignorant(x): “x is ignorant” vain(x): “x is vain” Universe: The set of all people. • No professors are vain • It is not the case that there is an x such that x is professor and x is vain. • For all people x, if x is a professor then x not vain.

20. Exercise professor(x): “x is a professor” ignorant(x): “x is ignorant” vain(x): “x is vain” Universe: The set of all people. • No professors are vain • It is not the case that there is an x such that x is professor and x is vain. • For all people x, if x is a professor then x not vain.

21. Precedence of Quantifiers • The quantifiers and have higher precedence then all logical operators from propositional calculus. • e.g. is the disjunction of .

22. Quantifiers with Restricted Domain

23. Quantifiers with Restricted Domain

24. Quantifiers with Restricted Domain

25. Quantifiers with Restricted Domain

26. Nested Quantifiers • “For all , there exists a such that”. • Example: • where and are integers

27. Nested Quantifiers • “For all , there exists a such that”. • Example: • where and are integers • There exists an x such that for all , is true” • Example: • THINK QUANTIFICATION AS LOOPS

28. Meanings of multiple quantifiers Suppose = “x likes y.” Domain of x: {St1, St2}; Domain of y: {DS, Calculus} • true for all x, y pairs. • true for at least one x, y pair. • For every value of x we can find a (possibly different) y so that P(x,y) is true. • There is at least one x for which P(x,y) is always true.

29. Meanings of multiple quantifiers Suppose = “x likes y.” Domain of x: {St1, St2}; Domain of y: {DS, Calculus} • true for all x, y pairs. • true for at least one x, y pair. • For every value of x we can find a (possibly different) y so that P(x,y) is true. • There is at least one x for which P(x,y) is always true.

30. Meanings of multiple quantifiers Suppose = “x likes y.” Domain of x: {St1, St2}; Domain of y: {DS, Calculus} • true for all x, y pairs. • true for at least one x, y pair. • For every value of x we can find a (possibly different) y so that P(x,y) is true. • There is at least one x for which P(x,y) is always true.

31. Meanings of multiple quantifiers Suppose = “x likes y.” Domain of x: {St1, St2}; Domain of y: {DS, Calculus} • true for all x, y pairs. • true for at least one x, y pair. • For every value of x we can find a (possibly different) y so that P(x,y) is true. • There is at least one x for which P(x,y) is always true.

32. Quantification order is not commutative

33. Example Domain: Real numbers • True/False??? • For all real numbers x and for all real numbers y there is a real number z such that . • True • True/False??? • There is a real number z such that for all real numbers x and for all real numbers y it is true that . • False

34. Example Domain: Real numbers • True/False??? • For all real numbers x and for all real numbers y there is a real number z such that . • True • True/False??? • There is a real number z such that for all real numbers x and for all real numbers y it is true that . • False

35. Example Domain: Real numbers • True/False??? • For all real numbers x and for all real numbers y there is a real number z such that . • True • True/False??? • There is a real number z such that for all real numbers x and for all real numbers y it is true that . • False

36. Example Domain: Real numbers • True/False??? • For all real numbers x and for all real numbers y there is a real number z such that . • True • True/False??? • There is a real number z such that for all real numbers x and for all real numbers y it is true that . • False

37. From Nested Quantifiers to English • F (a, b): “a and b are friends” • Domain: All students in COMSATS. • There is a student x such that for all students y and all students z other than y, if x and y are friends and x and z are friends, then y and z are not friends. • There is a student none of whose friends are also friends with each other.

38. From Nested Quantifiers to English • F (a, b): “a and b are friends” • Domain: All students in COMSATS. • There is a student x such that for all students y and all students z other than y, if x and y are friends and x and z are friends, then y and z are not friends. • There is a student none of whose friends are also friends with each other.

39. From English to Nested Quantifiers • "If a person is female and is a parent, then this person is someone's mother“ • For every person x , if person x is female and person x is a parent, then there exists a person y such that person x is the mother of person y.“ • F(x): “x is female” • P(x): “x is a parent“ • M(x, y) : “x is the mother of y”

40. From English to Nested Quantifiers • "If a person is female and is a parent, then this person is someone's mother“ • For every person x , if person x is female and person x is a parent, then there exists a person y such that person x is the mother of person y.“ • F(x): “x is female” • P(x): “x is a parent“ • M(x, y) : “x is the mother of y”

41. The sum of two positive integers is always positive. • What is domain above? • Integers • If domain is “+ve integers”

42. The sum of two positive integers is always positive. • What is domain above? • Integers • If domain is “+ve integers”

43. The sum of two positive integers is always positive. • What is domain above? • Integers • If domain is “+ve integers”

44. Everyone has exactly one best friend • For every person x , person x has exactly one best friend. • B(x,y): “x has best friend y” • Exactly one best friend ????

45. Everyone has exactly one best friend • For every person x , person x has exactly one best friend. • B(x,y): “x has best friend y” • Exactly one best friend ????

46. Everyone has exactly one best friend • For every person x , person x has exactly one best friend. • B(x,y): “x has best friend y” • Exactly one best friend ????

47. Everyone has exactly one best friend • For every person x , person x has exactly one best friend. • B(x,y): “x has best friend y” • Exactly one best friend ????

48. Everyone has exactly one best friend • For every person x , person x has exactly one best friend. • B(x,y): “x has best friend y” • Exactly one best friend ????

49. Everyone has exactly one best friend • For every person x , person x has exactly one best friend. • B(x,y): “x has best friend y” • Exactly one best friend ????

50. There is a woman who has taken a flight on every airline in the world. • Domains: people airlines flights • W(x): x is a woman • F(x, f): x has taken flight f • A(f, a): flight f belongs to airline a