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Ch 1.3: Quantifiers

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?

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Ch 1.3: Quantifiers

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  1. 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.

  2. 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.

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

  4. 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.

  5. 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.

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

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

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

  9. 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

  10. Equivalence • Definition: Two quantified sentences are equivalentfor 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:

  11. Equivalence • Example: The following quantified sentences are equivalent.

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

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

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

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

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

  17. Unique existence quantifier • Definition:

  18. Examples

  19. Uniqueness equivalence & negation

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

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