1. Section 1.5 and 1.6 Predicates and Quantifiers

2. Vocabulary Predicate Domain Universal Quantifier Existential Quantifier Counterexample Free variable Bound variable

3. Activity 1 Define: –Proposition A statement that can be identified as true or false but not both. –Predicate A statement whose truth value is a function of one or more variables.

4. In the first two weeks So far we have looked at : –the concept of proposition statements –Several common operators/connectives ¬ 

5. But that isn’t always enough We talked about the fact that not all sentences are statements: He is the vice president. x+y>10 The true or false nature of these sentences depends on which values are bound into the variables (he, x, and y)

6. But that isn’t always enough Similarly, suppose we know: –Every computer connected to the university network is functioning properly. No rules of propositional logic allow us to conclude that –This laptop is functioning properly

7. We need more Thus, the next couple of sections introduce another form (a more powerful form) of logic that helps us with these deficiencies. This type of logic is called predicate logic and is built on the idea of allowing us to write statements that indicate a quantity.

8. The Logic of Quantified Statements A predicate is a sentence that contains a finite number of variables and becomes a statement when specific values are substituted for the variables. –“He is the vice president” is a sentence –It becomes a predicate statement when he=Joe Biden or he=Paul Andersen or even he=Ben Schafer

9. The Logic of Quantified Statements The domain of a predicate variable is the set of all values that may be substituted in place of the variable. What is the domain of “he” –The domain for he in the sentence “He is the vice president” is the set of all humans.

10. The Logic of Quantified Statements If P(x) is a predicate and x has domain D, the truth set of P(x) is the set of all elements of D that make P(x) true when they are substituted for x. The truth set of P(x) is denoted: { x  D | P(x) }

11. Activity #2 Let P(x) denote the statement “x+2>5” –What are the truth values of P(4), P(2) and P(3)? –P(4) means “4+2>5” and that is true –P(2) means “2+2>5” and that is false –P(3) means “3+2>5” and that is false

12. Example Let P(x) denote the statement “x+2>5” –Let the domain of x be Z (the set of all integers) –Then the truth set is {x  Z | x>3 }

13. Questions??

14. Quantifiers Quantifiers are words that refer to quantities such as “some” or “all” and tell for how many elements a given predicate is true. In predicate logic we focus on two quantifiers:

15. The Universal Quantifier The symbol  is read as “for all” Let Q(x) be a predicate and D the domain of x. A universal statement is a statement of the form –  x  D, Q(x) For all x who are human beings, x is mortal.  human beings h, h is vice president.

16. The Universal Quantifier A universal statement is a statement of the form: –  x  D, Q(x) This statement is defined to be true if and only if Q(x) is true for every x. This statement is defined to be false if for at least one value of x exists such that Q(x) is false. X is then called a counterexample.

17. The Universal Quantifier Assume that x1, x2, x3… xn are EVERY x in the set D. Then:  x  D, Q(x) Is the same as saying –Q(x1)  Q(x2)  Q(x3)  …  Q(xn)

18. The Existential Quantifier The symbol  is read as “there exists” Let Q(x) be a predicate and D the domain of x. An existential statement is a statement of the form –  x  D, Q(x) There exists an integer x such that x*2 = x. There exists an integer x such that x+1 = x.

19. The Existential Quantifier An existential statement is a statement of the form: –  x  D, Q(x) This statement is defined to be true if and only if Q(x) is true for at least one value of x in D. This statement is defined to be false if it is false for all x in D.

20. The Existential Quantifier Assume that x1, x2, x3… xn are EVERY x in the set D. Then: –  x  D, Q(x) Is the same as saying –Q(x1)  Q(x2)  Q(x3)  …  Q(xn)

21. Activity #3 How would you prove that  x  D, Q(x) is true  x  D, Q(x) is false  x  D, Q(x) is true  x  D, Q(x) is false

22. Activity #4 Write down a true universal statement Write down a false universal statement Write down a true existential statement Write down a false existential statement

25. Activity #5 Assume: –the domain consists of integers –O(x) is “x is odd” –L(x) is “x < 10” –G(x) is “x>9”

26. Activity #5 Given the prior assumptions, what is the truth value of the following statements. 1.  x [ O(x) ] 2.  x [L(x)  O(x) ] 3.  x [L(x)  ¬ G(x) ] 4.  x [L(x)  G(x)] 5.  x [L(x)  G(x)

27. Activity #6 Assume: –the domain consists of integers –A(x) is “x<5” –B(x) is “x<7”

28. Activity #6 Given the prior assumptions, what is the truth value of the following statements. 1.  x [ A(x) ] 2.  x [ A(x)  B(x)] 3.  x [ A(x)  B(x)] 4.  x [ A(x)  B(x) ] 5.  x [ B(x)  A(x) ]