Ch 1 3 Quantifiers Open sentences or predicates
- Slides: 20
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: 2 x + y = 7 Let P(x, y) = “ 2 x + 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.
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 “ 2 x + 5 = 7” and Q(x) be “x = 1”, universe = R. Then P(x) and Q(x) are equivalent.
Universal & existential quantifiers Definitions: Given an open sentence P(x),
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.
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.
Examples Example: Translate “Chickens with jobs ride the bus” into a symbolic sentence with quantifiers. Universe = all animals
Examples Example: Translate “Some chickens with jobs have a car” into a symbolic sentence with quantifiers. Universe = all animals
Examples Example: Translate “A function f has an inverse if different inputs give different outputs” into a symbolic sentence with quantifiers. Universe = R
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
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:
Equivalence Example: The following quantified sentences are equivalent.
Negation of Quantifiers Theorem: For the open sentence A(x), �
Examples Example: Negate “Chickens with jobs ride the bus” (Universe = all animals)
Examples Example: Negate “Some chickens with jobs have a car” (Universe = all animals)
Examples Example: Negate “A function f has an inverse if different inputs give different outputs. ” (Universe = R)
Examples Example: Negate “For every natural number there is a real number greater than the natural number” (Universe = R)
Unique existence quantifier Definition:
Examples
Uniqueness equivalence & negation
Homework Read Ch 1. 3 Do 24(1 a-j, 2 a-j, 4 a-c, f, g, 5 a-c, f, 6 a-d, g, 10)
- Predicates and quantifiers examples
- Applications of predicates and quantifiers
- Unit 1 subjects predicates and sentences
- Predicate
- Open propositions and quantifiers
- Open innovation open science open to the world
- Transition and transfer predicates
- What is a compound predicate
- Com_min algorithm
- Simple predicate examples
- Transition and transfer predicates
- Complete subject and simple predicate
- Compound predicate
- Predicate and predicator
- Draw a line between the complete subject and predicate
- Predicate song
- Lesson 3 subjects and predicates
- Subjects predicates
- Computable predicates in ai
- Compound quantifiers
- Quantifiers to make comparisons