1. RMIT University; Taylor's College This is a story about four people named Everybody, Somebody, Anybody and Nobody. There was an important job to be done and Everybody was sure that Somebody would do it. Anybody could have done it, but Nobody did it. Somebody got angry about that, because it was Everybody's job. Everybody thought that Anybody could do it, but Nobody realized that Everybody wouldn't do it. It ended up that Everybody blamed Somebody when Nobody did what Anybody could have done.

2. RMIT University; Taylor's College Lecture 8  To apply quantifiers on predicates  To apply the de Generalized de Morgan’s Laws  To determine the truth value of predicates involving combination of two quantifiers

3. RMIT University; Taylor's College Universal Quantifiers: All

4. RMIT University; Taylor's College Universal Quantifiers  Let P(x) represent the predicate: “Student x has brains.”  Then “every student has brains” becomes: or more simply,  When we don’t need to specify the domain, this becomes:

5. RMIT University; Taylor's College Universal Quantifiers Exercise:

6. RMIT University; Taylor's College

7. Existential Quantifiers: There Exist  How do we say, “There exists a student with brains”?  How do we say, “Some students have brains”?  In logic, “some” means “at least one”.  This is how we apply one of the quantifiers to a predicate.

8. RMIT University; Taylor's College Existential Quantifiers Exercise:

9. Existential Quantifiers: There Exist RMIT University; Taylor's College

10. Quantifiers  A statement involving predicates whose variables are all properly quantified becomes a proposition (provided that the domain is known).

11. RMIT University; Taylor's College Quantifiers  Example: D = Z = {…, -3, -2, -1, 0, 1, 2, 3, …} P(x) means “x is positive” Which are true?  This means “for all x, P(x)” which means “for all x, x is positive.” This is false. A counterexample is x = 0 (or x = -1, etc) A counterexample is an example which proves that a universally quantified statement is false.  This means “there exists x such that P(x) is true” which means “there exists x such that x is positive”. This is true. For example, x = 1.

12. RMIT University; Taylor's College Quantifiers  It’s more likely that is true than that is true.  In fact, provided that the domain D is nonempty,

13. RMIT University; Taylor's College The Generalized de Morgan Laws  What happens if we negate an expression involving predicates and quantifiers?  The Generalized de Morgan Laws  Examples: 1.“It’s not true that all food is delicious” is the same as “there exists some food which is not delicious.” 2.“It’s not true that some dogs bite” is the same as “there aren’t any dogs who bite” or equivalently “all dogs don’t bite”.

14. RMIT University; Taylor's College Combining Quantifiers  A predicate can have numerous variables, each of which may be quantified.  Example  It can be difficult to interpret expressions involving 3 or more quantifiers. Negation

15. RMIT University; Taylor's College Combination of Quantifiers  Having two quantifiers is a lot easier.  What do these mean? Which are true in any given situation?  This depends on how P(x, y) is defined, and on what set is chosen as the domain of interpretation D.

16. RMIT University; Taylor's College http://newsimg.bbc.co.uk/media/images/46343000/gif/_46343078_height_world_leaders_466.gif Accessed 29 th September 2009 P(x, y): x is taller than y Everyone is taller than everyone. There exists a person who is taller than everyone. Everyone is taller than someone. Everyone is shorter than someone. There is someone who is shorter than everyone. There is someone who is taller than another person.

17. RMIT University; Taylor's College Example  Suppose P(x, y) means x ≥ y. Let D be the set N \ {0} = {1, 2, 3, … }.  Which of the six predicate formulae given in the previous slide are true?  The discussion is in the following slides.

18. RMIT University; Taylor's College  For all x and (for all) y, P(x, y) is true.  This says that no matter which numbers x and y we choose from N \ {0}, it will always happen that x ≥ y.  Is this true?  No. For example, x = 1 and y = 2.

19. RMIT University; Taylor's College  For all x there exists y such that x ≥ y.  Here y can depend on x.  A different choice of x may lead to a different value of y.  Is this true?  Yes. For example, given x we can take y = x. Then x ≥ y. Or, we could take y = 1 when x = 1, and take y = x – 1 for all other values of x. Or we could take y = 1 always.

20. RMIT University; Taylor's College  There exists x such that for all y, x ≥ y.  This says that x is a constant, and every choice of y makes x ≥ y.  Is this true?  No. There is no such constant. (It would have to be the biggest integer – the largest element of N \ {0}. But this set has no largest element.)

21. RMIT University; Taylor's College  There exists x and there exists y such that x ≥ y.  Is this true?  Yes. For example, take x = 2 and y = 1.

22. RMIT University; Taylor's College  For all y there exists x such that x ≥ y.  This says that for every choice of y it ’ s possible to find an x which is ≥ y.  Is this true?  Yes. For example, put x = y + 1 (or take x = y)

23. RMIT University; Taylor's College  There exists y such that for all x, x ≥ y.  This says that there is a constant y which is less than or equal to all values of x.  Is this true?  Yes: y = 1 has this property. It ’ s the smallest element of the set.

24. RMIT University; Taylor's College Exercises 1.How do the results change if D changes to the finite set {1, 2, 3, 4, 5}? 2.How do the results change if D changes from N \ {0} to Z? 3.Is there a domain for which all six formulae are true? 4.How do the results change if P(x, y) changes to “x > y”?