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Chapter 2: Problem 11

\(\forall a \in \mathbf{Z},(a-1) / a\) is not an integer.

Short Answer

Expert verified
To show that for any integer \(a\), the fraction \((a-1) / a\) is not an integer, we consider whether \((a-1)\) is divisible by \(a\). If there existed an integer \(k\) such that \(a-1 = ak\), then we would have \(k = \frac{a-1}{a}\), which would mean that \(\frac{a-1}{a}\) is an integer. However, this contradicts the given statement that \((a-1) / a\) is not an integer. Thus, there is no such \(k\), and we conclude that \((a-1) / a\) is not an integer for all \(a \in \mathbf{Z}\).

Step by step solution

01

Identify the Given Information

We are given the set of integers \(\mathbf{Z}\) and a statement that needs to be proved for all integers \(a\): \((a-1) / a\) is not an integer.
02

Test Divisibility

In order for \((a-1) / a\) to be an integer, the numerator (\(a-1\)) must be divisible by the denominator (\(a\)) with no remainder. We can say that \(a-1\) is divisible by \(a\) if there exists an integer \(k\) such that \(a-1 = ak\).
03

Find the Counterexample

Suppose that there exists an integer \(k\) such that \(a-1 = ak\). We can rearrange the equation to find the value of \(k\): \(k = \frac{a-1}{a}\) Since \(k\) is an integer, \(\frac{a-1}{a}\) must also be an integer. However, this contradicts the given statement in which \((a-1) / a\) is, in fact, not an integer. Therefore, there cannot be such a \(k\), and we can conclude that \((a-1) / a\) is not an integer for all \(a \in \mathbf{Z}\).

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Most popular questions from this chapter

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. $$ \forall \text { odd integers } n, \exists \text { an integer } k \text { such that } n=2 k+1 \text {. } $$

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.

Indicate whether the arguments in \(21-26\) are valid or invalid. Support your answers by drawing diagrams. No college cafeteria food is good. No good food is wasted. \(\therefore\) No college cafeteria food is wasted.

Which of the following is a negation for "All dogs are loyal"? More than one answer may be correct. a. All dogs are disloyal. b. No dogs are loyal. c. Some dogs are disloyal. d. Some dogs are loyal. e. There is a disloyal animal that is not a dog. f. There is a dog that is disloyal. g. No animals that are not dogs are loyal. \(\mathrm{h}\). Some animals that are not dogs are loyal.

Let \(D\) be the set of all students at your school, and let \(M(s)\) be "s is a math major," let \(C(s)\) be " \(s\) is a computer science student," and let \(E(s)\) be " \(s\) is an engineering student." Express each of the following statements using quantifiers, variables, and the predicates \(M(s), C(s)\), and \(E(s)\). a. There is an engineering student who is a math major. b. Every computer science student is an engineering student. c. No computer science students are engineering students. d. Some computer science students are also math majors. e. Some computer science students are engineering students and some are not.
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