1. Discrete Structures – CNS 2300
Text
Discrete Mathematics and Its Applications
Kenneth H. Rosen (5th Edition)
Chapter 1
The Foundations: Logic and Proof, Sets and Functions

2. Predicates & Quantifiers
Section
Predicates & Quantifiers

3. Open Statement
x >8
p < q -5
x = y + 6
Neither true nor false

4. Propositional Functions
x > 8
x is greater than 8. x
subject
is greater than predicate

5. Propositional Functionsx
subject
is greater than predicate
P(x)
propositional function P at x

6. Propositional Functions
There are two ways a propositional function
P(x) “x > 8”
can become true or false (a proposition)

7. P(x) “x > 8” P(5) “5 > 8” FALSE P(12) “12 > 8” TRUE
1. The variable may be given a value
P(5)
“5 > 8”
FALSE
P(12)
“12 > 8”
TRUE

8. Propositional Functions (two variables)
Let Q(x,y) be the statement “x is the capital of y.”
What are the truth values of:
Q(Carson City, Nevada)
Q(Detroit, Michigan)
Q(Nebraska, Lincoln)

9. P(x) “x > 8” Quantifiers 2. A propositional function
can become true or false (a proposition) by using quantifiers.

10. Universal Quantification
A Universal Quantifier states that P(x) is true for all values of x in the universe of discourse.
states the universal quantification of P(x)

11. Every student in this class has had CNS1250 or its equivalent.
Let P(x) denote the statement “x has had CNS1250 or its equivalent.”
represents the statement, where the universe of discourse for x is this class.

12. True

13. False

14. False

15. False

16. If S(x) represented the statement “x is in this class” and
P(x) denotes the statement “x has had CSIS1010 or its equivalent.”
also represents the statement.

17. Universe of Discourse
It is important to note that the universe of discourse must be defined before the logical value of a propositional function may be discussed.

18. No student spends more than five hours every weekday in class
Example:
Let P(x) be the statement “ x spends more than five hours every weekday in class,” where the universe of discourse for x is the set of students. Express in English.
No student spends more than five hours every weekday in class

19. Existential Quantification
States that P(x) is true for some value of x in the universe of discourse.
states the existential quantification of P(x)

20. Existential Quantification
There is at least one person in this class who has completed CNS2400.
Let P(x) represent the statement “x has completed CNS2400” where the universe of discourse is this class.
then represents the above statement.

21. True

22. True

23. True

24. False

25. Example:
Let P(x) be the statement “ x spends more than five hours every weekday in class,” where the universe of discourse for x is the set of students. Express in English.
There is some student (maybe more than one) who does not spend more than five hours every weekday in class.

26. Quantifiers FALSE TRUE There is some x for which P(x) is false
P(x) must be true for every x.
P(x) must be false for every x.
There is some x for which P(x) is true.

27. Binding Variables
When a quantifier is used on the variable x or when we assign a value to this variable, we say that this occurrence of the variable is bound.
Variable food, now bound with value fly.

28. Quantifiers with Multiple Variables

29. Multiple Variables

30. Take care in reading these.
Multiple Variables
Take care in reading these.

31. Example:
Let P(x,y) be the statement “x has taken class y,” where the universe of discourse for x is the set of all students in this class and for y is the set of all CNS courses at UVSC.

32. For every student in this class, there is a CNS class that student has taken.
Students in this class
CNS Courses

33. There is a CNS course that every student in this class has taken.
Students in this class
CNS Courses

34. Example: page 33, # 6
Let P(x,y) be the statement “x has taken class y,” where the universe of discourse for x is the set of all students in this class and for y is the set of all CNS courses at UVSC.

35. For every (pick any one you want) CNS course, there is a student in this class who has taken the course.
Students in this class
CNS Courses

36. There is a student in this class who has taken every CNS class.
Students in this class
CNS Courses

37. Negating Quantifiers

38. Let P(x) be the statement “the word x contains the letter m.”
Example
Let P(x) be the statement “the word x contains the letter m.”
There is no word that contains the letter m.
For any word you pick, that word does not contain the letter m.

39. Let P(x) be the statement “the word x contains the letter m.”
Example
Let P(x) be the statement “the word x contains the letter m.”
Not every word contains the letter m.
There is a word that does not contain the letter m.

40. Problems from the text
Homework will not be collected. However, you should do enough problems to feel comfortable with the concepts. For these sections the following problems are suggested.
Pages 40-42
1,3,7,9,13,19,21,29
Pages 51-54
1,3 7,9,15,27

41. finished