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The equivalence of Exercise 43 says that if it is false that every element of the domain has property A, then some element of the domain fails to have property A, and vice versa. The element that fails to have property A is called a counterexample to the assertion that every element has property A. Thus a counter- example to the assertion (Vx)(x is odd) in the domain of integers is the number 10, an even integer. (Of course, there are lots of other counterex- amples to this assertion.) Find counterexamples in the domain of integers to the following assertions. (An integer x>1 is prime if the only factors of x are 1 and x.) a. (Vx)(x is negative) b. (Vx)(x is the sum of even integers) c. (Vx)(x is prime →x is odd) d. (Vx)(x prime →(-1)* = -1) e. (Vx)(x prime →2* – 1 is prime)

The equivalence of Exercise 43 says that if it is false that every element of the domain has property A, then some element of the domain fails to have property A, and vice versa. The element that fails to have property A is called a counterexample to the assertion that every element has property A. Thus a counter- example to the assertion (Vx)(x is odd) in the domain of integers is the number 10, an even integer. (Of course, there are lots of other counterex- amples to this assertion.) Find counterexamples in the domain of integers to the following assertions. (An integer x>1 is prime if the only factors of x are 1 and x.) a. (Vx)(x is negative) b. (Vx)(x is the sum of even integers) c. (Vx)(x is prime →x is odd) d. (Vx)(x prime →(-1)* = -1) e. (Vx)(x prime →2* – 1 is prime)

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter2: Systems Of Linear Equations
Section2.4: Applications
Problem 33EQ
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The equivalence of Exercise 43 says that if it is false that every element of the domain has property A, then some element of the domain fails to have property A, and vice versa. The element that fails to have property A is called a counterexample to the assertion that every element has property A. Thus a counter- example to the assertion (Vx)(x is odd) in the domain of integers is the number 10, an even integer. (Of course, there are lots of other counterex- amples to this assertion.) Find counterexamples in the domain of integers to the following assertions. (An integer x>1 is prime if the only factors of x are 1 and x.) a. (Vx)(x is negative) b. (Vx)(x is the sum of even integers) c. (Vx)(x is prime →x is odd) d. (Vx)(x prime →(-1)* = -1) e. (Vx)(x prime →2* – 1 is prime)
Transcribed Image Text:The equivalence of Exercise 43 says that if it is false that every element of the domain has property A, then some element of the domain fails to have property A, and vice versa. The element that fails to have property A is called a counterexample to the assertion that every element has property A. Thus a counter- example to the assertion (Vx)(x is odd) in the domain of integers is the number 10, an even integer. (Of course, there are lots of other counterex- amples to this assertion.) Find counterexamples in the domain of integers to the following assertions. (An integer x>1 is prime if the only factors of x are 1 and x.) a. (Vx)(x is negative) b. (Vx)(x is the sum of even integers) c. (Vx)(x is prime →x is odd) d. (Vx)(x prime →(-1)* = -1) e. (Vx)(x prime →2* – 1 is prime)
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