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Chapter 2: Problem 17

In each of 14-19, (a) rewrite the statement in English without using the symbol \(\forall\) or \(\exists\) but expressing your answer as simply as possible, and (b) write a negation for the statement. $$ \forall r \in \mathbf{Q} . \exists \text { integers } a \text { and } b \text { such that } r=a / b \text {. } $$

Short Answer

Expert verified
Every rational number r can be expressed as a fraction a/b, where a and b are integers. The negation is: There is a rational number r for which it is not possible to find integers a and b such that r is equal to a divided by b.

Step by step solution

01

Identify the variables and their domains

In our given statement, we have two variables: r, which belongs to the set of rational numbers (Q), and a and b, which are both integers.
02

Explain the quantifiers 鈭� and 鈭� and their purpose

In the given statement, the quantifier 鈭� is used to express the idea that the statement that follows is true for all elements in the specified domain (rational numbers, in this case). The quantifier 鈭� is used to express the existence of certain elements (integers a and b) for which the following statement is true.
03

Rewrite the statement in English without using the symbol 鈭� or 鈭�

The given statement translates to the following simple English sentence: "Every rational number r can be expressed as a fraction a/b, where a and b are integers." #b) Write a negation for the statement#
04

Negate the quantifiers

To negate the original statement, we need to change the quantifiers from 鈭� to 鈭� and from 鈭� to 鈭�. The new statement will now start with 鈭� instead of 鈭� and 鈭� will replace 鈭�.
05

Negate the original statement

Applying the above rule, the negated statement is: $$\exists r \in \mathbf{Q} . \forall \text{ integers } a \text{ and } b \text{, it is not true that } r=a / b \text{.}$$
06

Rewrite the negation in English

The negation of the original statement translates to the following simple English sentence: "There is a rational number r for which it is not possible to find integers a and b such that r is equal to a divided by b."

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Rational Numbers
Rational numbers are a fundamental concept in mathematics, especially within discrete mathematics. A rational number is any number that can be expressed as a fraction \frac{a}{b} where 'a' and 'b' are integers, and 'b' is not zero. This definition includes all integers, since any integer 'z' can be written as \frac{z}{1}, and any fraction with integer numerator and denominator.

Rational numbers are denoted by the symbol \( \mathbf{Q} \). They include numbers such as \( \frac{1}{2} \) (one half), \( \frac{-3}{4} \) (negative three-fourths), and \(5 \) (which can be written as \( \frac{5}{1} \) ). The ability to express any rational number as a fraction is a powerful concept because it allows for precise calculations and comparisons between different quantities.
The Negation of Quantified Statements
Understanding quantified statements and their negation is crucial in the realms of logic and discrete mathematics. Quantified statements include universal quantifiers ('for all,' denoted by \( \forall \) ) and existential quantifiers ('there exists,' denoted by \( \exists \)). The negation of a statement involves switching these quantifiers and altering the statement to reflect what is not true.

To negate a universally quantified statement (\( \forall \)), you replace it with an existential quantifier (\( \exists \)), and vice versa. Once the quantifiers are switched, the following proposition is also negated. For example, negating 'Every student in the class is wearing glasses' would result in 'There exists a student in the class who is not wearing glasses.' The process of negation is crucial in constructing proofs and understanding logical equivalences in mathematics.
Expression of Rational Numbers
Expressing rational numbers is about writing them in the form \( \frac{a}{b} \) where 'a' and 'b' are integers and 'b' is not zero. This form is known as a fraction. However, it's important to note that not just any two integers will do; 'b' must not be zero because division by zero is undefined. Additionally, when expressing rational numbers in their lowest terms, 'a' and 'b' should be co-prime, meaning they share no common factors other than one.

Clever manipulation of these fractions, such as finding common denominators, allows us to add, subtract, multiply, and divide them in an orderly fashion. Furthermore, rational numbers can be converted into decimal numbers, though not all decimals are rational (consider non-repeating, non-terminating decimals). Discrete mathematics often involves working with rational numbers to solve problems related to number theory, computer science, and logic.

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Most popular questions from this chapter

Give an example to show that a universal conditional statement is not logically equivalent to its inverse. Passing a comprehensive exam is a necessary condition for obtaining a master's degree.

If the square of an integer is odd, then the integer is odd.

Some of the arguments in 7-18 are valid by universal modus ponens or universal modus tollens; others are invalid and exhibit the converse or the inverse error. State which are valid and which are invalid. Justify your answers. All healthy pcople cat an apple a day. Herbert is not a healthy person. Herbert does not eat an apple a day.

In informal speech most sentences of the form "There is ___ every ___ are intended to be understood as tial quantifier there is comes before the universal quantifier every. Note that this interpretation applies to the following well-known sentences. Rewrite them using quantifiers and variables. a. There is a sucker born every minute. b. There is a time for every purpose under heaven.

In exercises \(28-33\), reorder the premises in each of the arguments to show that the conclusion follows as a valid consequence from the premises. It may be helpful to rewrite the statements in ifthen form and replace some statements by their contrapositives. Exercises 28-30 refer to the kinds of Tarski worlds discussed in Example 2.1.12 and 2.3.1. Exercises 31 and 32 are adapted from Symbolic Logic by Lewis Carroll.* 1\. I trust every animal that belongs to me. 2\. Dogs gnaw bones. 3\. I admit no animals into my study unless they will beg when told to do so. 4\. All the animals in the yard are mine. 5\. I admit every animal that I trust into my study. 6\. The only animals that are really willing to beg when told to do so are dogs. \(\therefore\) All the animals in the yard gnaw bones.
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