/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 2 For each of the following, use a... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 2

For each of the following, use a counterexample to show that the statement is false. Then write the negation of the statement in English, without using symbols for quantifiers. * (a) \((\forall m \in \mathbb{Z})\left(m^{2}\right.\) is even \() .\) (b) \((\forall x \in \mathbb{R})\left(x^{2}>0\right)\) (c) For each real number \(x, \sqrt{x} \in \mathbb{R}\) (d) \((\forall m \in \mathbb{Z})\left(\frac{m}{3} \in \mathbb{Z}\right)\). (e) \((\forall a \in \mathbb{Z})\left(\sqrt{a^{2}}=a\right)\). "(f) \((\forall x \in \mathbb{R})\left(\tan ^{2} x+1=\sec ^{2} x\right)\)

Short Answer

Expert verified
a) Counterexample: m = 1, \(1^2=1\) is not even. Negation in English: "There exists an integer whose square is not even." b) Counterexample: x = 0, \(0^2=0\) is not greater than 0. Negation in English: "There exists a real number whose square is not greater than 0." c) Counterexample: x = -1, \(\sqrt{-1}=i\) is not a real number. Negation in English: "There exists a real number whose square root is not a real number." d) Counterexample: m = 1, \(\frac{1}{3}\) is not an integer. Negation in English: "There exists an integer whose division by 3 is not an integer." e) Counterexample: a = -1, \(\sqrt{(-1)^2}=1\) is not equal to -1. Negation in English: "There exists an integer for which the square root of its square is not equal to the integer itself." f) The given statement is true, so there is no counterexample or negation in English.

Step by step solution

01

Find a counterexample for the statement \( (\forall m \in \mathbb{Z})\left(m^2\right.\) is even \()\)

We can find a counterexample by choosing an integer for which its square is not even. \( m = 1 \implies 1^2 = 1 \) is not even, since it is not divisible by 2. Therefore, the statement is false. Negation in English: "There exists an integer whose square is not even." b)
02

Find a counterexample for the statement \( (\forall x \in \mathbb{R})\left(x^{2}>0\right) \)

We can find a counterexample by choosing a real number for which its square is not greater than 0. \( x = 0 \implies 0^2 = 0 \) is not greater than 0. Therefore, the statement is false. Negation in English: "There exists a real number whose square is not greater than 0." c)
03

Find a counterexample for the statement "For each real number x, \(\sqrt{x} \in \mathbb{R}\) "

We can find a counterexample by choosing a real number for which the square root of the number is not a real number. If x = -1, then \( \sqrt{-1} = i \), the imaginary unit, which is not a real number. Therefore, the statement is false. Negation in English: "There exists a real number whose square root is not a real number." d)
04

Find a counterexample for the statement \( (\forall m \in \mathbb{Z})\left(\frac{m}{3} \in \mathbb{Z}\right) \)

We can find a counterexample by choosing an integer for which its division by 3 is not an integer. If m = 1, then \( \frac{1}{3} \) is not an integer, since it is a fraction. Therefore, the statement is false. Negation in English: "There exists an integer whose division by 3 is not an integer." e)
05

Find a counterexample for the statement \( (\forall a \in \mathbb{Z})\left(\sqrt{a^{2}}=a\right) \)

We can find a counterexample by choosing an integer for which the square root of its square is not equal to itself. If a = -1, then \( \sqrt{(-1)^2} = \sqrt{1} = 1 \), which is not equal to -1. Therefore, the statement is false. Negation in English: "There exists an integer for which the square root of its square is not equal to the integer itself." f)
06

Find a counterexample for the statement \( (\forall x \in \mathbb{R})\left(\tan^{2} x+1=\sec^{2} x\right) \)

This statement corresponds to the well-known trigonometric identity \( \tan^2 x + 1 = \sec^2 x \). This identity holds true for all real numbers x such that the functions \(\tan x\) and \(\sec x\) are defined. We cannot find a counterexample for this statement since it is true. In this case, there is no negation in English, since the original statement is true.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Negation
Negation is an important concept in logic and mathematics. It is the process of denying or disproving a statement. A statement negation essentially flips the truth value of the original statement. If a statement is true, its negation is false, and vice versa.
Here's how you can understand negation in simple terms:
  • If a statement says "All cats are black," its negation would be "Not all cats are black," or "There exists a cat that is not black."
  • Negation is particularly useful when proving statements by contradiction or when we need to show that certain statements are false.
The goal of negation is to demonstrate the opposite of a given statement, often using a counterexample.
Mathematical Statements
In mathematics, statements, also known as propositions, are declarative sentences that are either true or false but not both. Understanding these statements is crucial for logical reasoning and proof.
Here are some characteristics of mathematical statements:
  • They must be clear and unambiguous. For instance, "2 is even" is a statement since it clearly holds true.
  • They include both universal statements, like "all integers are integer numbers," and existential statements, such as "there exists an even prime other than 2."
Mathematical statements can often be tested for validity using logical tools, such as a counterexample for checking falsehood.
Real Numbers
Real numbers are a fundamental mathematical concept encompassing all numbers that can be found on the number line. They include both rational and irrational numbers.
Some features of real numbers:
  • Rational numbers: These are numbers expressed as a fraction of two integers, like \( \frac{1}{2} \) or 2.
  • Irrational numbers: These numbers cannot be expressed as a simple fraction, like \( \pi \) or \( \sqrt{2} \).
  • Real numbers can be positive, negative, or zero, covering the entire range of values on the number line.
  • They are used to express measurements, like distance, area, and volume.
Real numbers are essential in numerous areas of mathematics and are the basis for calculus and analysis.
Integers
Integers form a set of numbers including positive whole numbers, negative whole numbers, and zero. They are an essential concept in mathematics, widely used in arithmetic and number theory.
Key aspects of integers:
  • Positive integers (1, 2, 3, ...), negative integers (-1, -2, -3, ...), and zero.
  • They do not include fractions or decimals. For example, 3 and -5 are integers, but \( \frac{1}{2} \) is not.
  • Integers are useful in counting, ordering, and various algebraic operations.
The set of integers is infinite in both the positive and negative directions, making it a vital part of any mathematical curriculum.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

For statements \(P\) and \(Q,\) use truth tables to determine if each of the following statements is a tautology, a contradiction, or neither. (a) \(\neg Q \vee(P \rightarrow Q)\) (c) \((Q \wedge P) \wedge(P \rightarrow \neg Q)\) (b) \(Q \wedge(P \wedge \neg Q)\) (d) \(\neg Q \rightarrow(P \wedge \neg P)\)

For statements \(P, Q,\) and \(R\) : (a) Show that \([(P \rightarrow Q) \wedge P] \rightarrow Q\) is a tautology. Note: In symbolic logic, this is an important logical argument form called modus ponens. (b) Show that \([(P \rightarrow Q) \wedge(Q \rightarrow R)] \rightarrow(P \rightarrow R)\) is a tautology. Note: In symbolic logic, this is an important logical argument form called Syllogism.

Use set builder notation to specify the following sets: (a) The set of all integers greater than or equal to 5. (b) The set of all even integers. \(\star(c)\) The set of all positive rational numbers. (d) The set of all real numbers greater than 1 and less than 7 . \star (e) The set of all real numbers whose square is greater than 10 .

Use the roster method to specify the truth set for each of the following open sentences. The universal set for each open sentence is the set of integers \(\mathbb{Z}\). (a) \(n+7=4\). (b) \(n^{2}=64\). (c) \(\sqrt{n} \in \mathbb{N}\) and \(n\) is less than 50 . (d) \(n\) is an odd integer that is greater than 2 and less than 14 . (e) \(n\) is an even integer that is greater than 10 .

Suppose that Daisy says, "If it does not rain, then I will play golf." Later in the day you come to know that it did rain but Daisy still played golf. Was Daisy's statement true or false? Support your conclusion.
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Decision Maths

Read Explanation

Discrete Mathematics

Read Explanation

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.