2. Ch 1.3: Quantifiers Open sentences, or predicates, are sentences that contain one or more variables. They do not have a truth value until variables are instantiated (replaced with particular value). Example: 2x + y = 7 Let P(x,y) = “2x + y = 7” What values of x, y make P(x,y) true? P(2,3) is true; P(1,5) is true; P(-2,11) is true; P(3,5) is false Truth set: Set of values which make open sentence true. Universe: Set of values that can be considered. Note: truth set may change when the universe changes.

3. Truth set, universe Example: Q(x) = “x^2 = 9” When universe is all reals R, the truth set is {-3, 3}. When universe is natural numbers N, truth set is{3}. Defn: Two open sentences P(x) and Q(x) are equivalent iff they have the same truth set, given a particular universe. Example: Let P(x) be “2x + 5 = 7” and Q(x) be “x = 1”, universe = R. Then P(x) and Q(x) are equivalent.

4. Universal & existential quantifiers Definitions: Given an open sentence P(x),

5. Universal & existential quantifiers Example: Translate “All apples have spots” into a symbolic sentence with quantifiers. Use A(x) = “x is an apple” and S(x) = “x has spots,” universe = all fruits.

6. Universal & existential quantifiers Example: Translate “Some apples have spots” into a symbolic sentence with quantifiers. Use A(x) = “x is an apple” and S(x) = “x has spots,” universe = all fruits.

7. Examples Example: Translate “Chickens with jobs ride the bus” into a symbolic sentence with quantifiers. Universe = all animals

8. Examples Example: Translate “Some chickens with jobs have a car” into a symbolic sentence with quantifiers. Universe = all animals

9. Examples Example: Translate “A function f has an inverse if different inputs give different outputs” into a symbolic sentence with quantifiers. Universe = R

10. Examples Example: Translate “For every natural number there is a real number greater than the natural number” into a symbolic sentence with quantifiers. Universe = R

11. Equivalence Definition: Two quantified sentences are equivalent for a particular universe if they have the same truth value in that universe. Definition: Two quantified sentences are equivalent iff they are equivalent in every universe. Example: The following quantified sentences are equivalent in N but not equivalent in R, hence they are not equivalent:

12. Equivalence Example: The following quantified sentences are equivalent.

13. Negation of Quantifiers Theorem: For the open sentence A(x), 

14. Examples Example: Negate “Chickens with jobs ride the bus” (Universe = all animals)

15. Examples Example: Negate “Some chickens with jobs have a car” (Universe = all animals)

16. Examples Example: Negate “A function f has an inverse if different inputs give different outputs.” (Universe = R)

17. Examples Example: Negate “For every natural number there is a real number greater than the natural number” (Universe = R)

18. Unique existence quantifier Definition:

19. Examples

20. Uniqueness equivalence & negation

21. Homework Read Ch 1.3 Do 24(1a-j,2a-j,4a-c,f,g,5a-c,f,6a-d,g,10)