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Let the domain of discourse be the set of all creatures. Define all propositional functions and symbols used in your proof. All dogs are loyal and empathic creatures. No selfish creature is loyal. Mochi is a dog. Therefore, Mochi is not selfish.

Let the domain of discourse be the set of all creatures. Define all propositional functions and symbols used in your proof. All dogs are loyal and empathic creatures. No selfish creature is loyal. Mochi is a dog. Therefore, Mochi is not selfish.

Elementary Geometry For College Students, 7e
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 16CT: Let P represent any statement. Classify as true or false. a P and P b P or P
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Let the domain of discourse be the set of all creatures. Define all propositional
functions and symbols used in your proof.

All dogs are loyal and empathic creatures. No selfish creature is loyal. Mochi is a
dog. Therefore, Mochi is not selfish.

Let the domain of discourse be the set of all creatures. Define all propositional functions and symbols used in your proof. All dogs are loyal and empathic creatures. No selfish creature is loyal. Mochi is a dog. Therefore, Mochi is not selfish.
Transcribed Image Text:Let the domain of discourse be the set of all creatures. Define all propositional functions and symbols used in your proof. All dogs are loyal and empathic creatures. No selfish creature is loyal. Mochi is a dog. Therefore, Mochi is not selfish.
4.3.2 Universally and Existentially Quantified Statements Definition: Let P be a propositional function with domain of discourse D. The statement "for every x, P(x)", is called a universally quantified statement The symbol V means "for every" and is called a universal quantifier. The universally quantified statement, Vx, P(x), is true if P(y) is true for every y in D and it is false if P(y) is false for at least oney in D. The statement "there exists x such that P(x)" is called an existentially quantified statement The symbol 3 means "there exists" and existential quantifier. The existentially quantified statement, 3x, P(x), is true if P(y) is true for at least one yin D and false if P(y) is false for every y in called an D. The variable x in the universally quantified statement or the existentially quantified statement is called a bound variable. A statement with a free variable is not a proposition while a statement with a bound variable is a proposition. Alternative ways to read V are "for all" and "for any". Alternative ways to read 3 are "for some" and "for at least one" To specify the domain of discourse, a universally quantified statement is written as: for every x in D, P(x) while an existentially quantified statement is written as: for some x in D, P(x). If a universally quantified statement is true then replacing the universal quantifier by an existentially quantifier will still yield a proposition that is true. If an existentially quantified statement is false then replacing the existential quantifier by a universal quantifier will still yield a proposition that is false. Definition: Let P(x) be a propositional function with domain of discourse D. The negation of a universally quantified statement is an existentially quantified statement as follows: ~(Vx, P(x))=3x, (~P(x)). The negation of an existentially quantified statement is a universally quantified statement as follows: ~(Ex, P(x))=Vx, (~P(x)).
Transcribed Image Text:4.3.2 Universally and Existentially Quantified Statements Definition: Let P be a propositional function with domain of discourse D. The statement "for every x, P(x)", is called a universally quantified statement The symbol V means "for every" and is called a universal quantifier. The universally quantified statement, Vx, P(x), is true if P(y) is true for every y in D and it is false if P(y) is false for at least oney in D. The statement "there exists x such that P(x)" is called an existentially quantified statement The symbol 3 means "there exists" and existential quantifier. The existentially quantified statement, 3x, P(x), is true if P(y) is true for at least one yin D and false if P(y) is false for every y in called an D. The variable x in the universally quantified statement or the existentially quantified statement is called a bound variable. A statement with a free variable is not a proposition while a statement with a bound variable is a proposition. Alternative ways to read V are "for all" and "for any". Alternative ways to read 3 are "for some" and "for at least one" To specify the domain of discourse, a universally quantified statement is written as: for every x in D, P(x) while an existentially quantified statement is written as: for some x in D, P(x). If a universally quantified statement is true then replacing the universal quantifier by an existentially quantifier will still yield a proposition that is true. If an existentially quantified statement is false then replacing the existential quantifier by a universal quantifier will still yield a proposition that is false. Definition: Let P(x) be a propositional function with domain of discourse D. The negation of a universally quantified statement is an existentially quantified statement as follows: ~(Vx, P(x))=3x, (~P(x)). The negation of an existentially quantified statement is a universally quantified statement as follows: ~(Ex, P(x))=Vx, (~P(x)).
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