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

Predicates & Quantifiers Section 1.3-1.4 Predicates & Quantifiers

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

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

Propositional Functions x subject is greater than 8. predicate P(x) propositional function P at x

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

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

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)

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

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)

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.

True

False

False

False

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.

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.

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

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)

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.

True

True

True

False

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.

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.

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.

Quantifiers with Multiple Variables

Multiple Variables

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

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.

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

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

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.

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

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

Negating Quantifiers

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.

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.

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

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