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Chapter 2: Problem 46

Translate each of these quantifications into English and determine its truth value. $$ \begin{array}{ll}{\text { a) } \exists x \in \mathbf{R}\left(x^{3}=-1\right)} & {\text { b) } \exists x \in \mathbf{Z}(x+1>x)} \\ {\text { c) } \forall x \in \mathbf{Z}(x-1 \in \mathbf{Z})} & {\text { d) } \forall x \in \mathbf{Z}\left(x^{2} \in \mathbf{Z}\right)}\end{array} $$

Short Answer

Expert verified
a) True, b) True, c) True, d) True

Step by step solution

01

Translate each quantification into English

Translate the given quantifications one by one.a) \( \exists x \in \mathbf{R}(x^3 = -1) \) means 'There exists an x in the set of real numbers such that x cubed equals -1.'b) \( \exists x \in \mathbf{Z}(x+1 > x) \) means 'There exists an x in the set of integers such that x plus 1 is greater than x.'c) \( \forall x \in \mathbf{Z}(x-1 \in \mathbf{Z}) \) means 'For all x in the set of integers, x minus 1 is an integer.'d) \( \forall x \in \mathbf{Z}(x^2 \in \mathbf{Z}) \) means 'For all x in the set of integers, x squared is an integer.'
02

Determine the truth value of a

Check if there is a real number x such that \( x^3 = -1 \).Solution: Yes, \( -1 \) is such a real number because \( (-1)^3 = -1 \).Truth value: True
03

Determine the truth value of b

Check if there is an integer x such that \( x + 1 > x \).Solution: This is true for any integer because adding 1 to x will always yield a number greater than x.Truth value: True
04

Determine the truth value of c

Check if for all integers x, \( x - 1 \) is an integer.Solution: Subtracting 1 from any integer still results in an integer.Truth value: True
05

Determine the truth value of d

Check if for all integers x, \( x^2 \) is an integer.Solution: Squaring any integer results in an integer.Truth value: True

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Key Concepts

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

Existential Quantifier
In logic, the existential quantifier is denoted as \( \exists \ \) and translates to 'there exists.' It is used to assert that there is at least one element in a particular set that satisfies a given property. For example, in the expression \( \exists x \in \mathbf{R}(x^3 = -1) \ \), it means 'there exists a real number x such that x cubed equals -1.' The statement is true if we can find at least one real number that makes the equation hold. In this case, x = -1 satisfies the condition, making the statement true. This quantifier is essential in identifying if any member of a set meets a specific criterion.
Universal Quantifier
The universal quantifier is denoted by \( \forall \ \) and means 'for all' or 'for every.' It is used to state that a property or predicate holds for all elements in a particular set. For example, in \( \forall x \in \mathbf{Z}(x^2 \in \mathbf{Z}) \ \), it asserts 'for all integers x, x squared is an integer.' This statement examines every single integer and verifies if squaring it remains within the set of integers. Since the square of any integer is always an integer, the statement is true. The universal quantifier is useful for proving properties or conditions that apply universally to all members of a set.
Mathematical Truth Values
Mathematical statements are either true or false. Understanding their truth values is critical. For instance, the statement \( \exists x \in \mathbf{Z}(x+1 > x) \ \) which means 'there exists an integer x such that x plus one is greater than x,' is true if such an integer exists. Since adding one to any integer always gives a larger number, this statement is true. Conversely, if no element satisfies the condition, the statement would be false. Hence, determining the truth value involves checking the validity of the given quantification. This process helps in validating or refuting mathematical claims.
Real Numbers
Real numbers include all rational and irrational numbers. They can be found on the number line and cover everything from negative to positive infinity, including fractions and decimals. For example, \( -1, 0, 1, \ \pi, \sqrt{2} \ \) are all real numbers. In the statement \( \exists x \in \mathbf{R}(x^3 = -1) \ \), we are searching within the set of real numbers to find an x that satisfies the equation. Since \( -1 \ \) is a real number and \( (-1)^3 = -1 \ \), the statement is true. Real numbers are pivotal in quantifying measurements and continuous quantities.
Integers
Integers are whole numbers that can be positive, negative, or zero. They do not include fractions or decimals. Examples of integers are \( ..., \ -3, -2, -1, 0, 1, 2, 3, ... \ \). When working with integers in logical statements, it's important to remember their properties. For example, in \( \forall x \in \mathbf{Z}(x-1 \in \mathbf{Z}) \ \), the statement asserts 'for all integers x, x minus 1 is also an integer.' Subtracting one from any integer will always result in another integer. Hence, the statement is true. Understanding integers and their behavior is crucial when dealing with discrete quantities and whole numbers in mathematics.

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Most popular questions from this chapter

Find \(\bigcup_{i=1}^{\infty} A_{i}\) and \(\bigcap_{i=1}^{\infty} A_{i}\) if for every positive integer \(i\) a) \(A_{i}=\\{i, i+1, i+2, \ldots\\}\) b) \(A_{i}=\\{0, i\\}\) c) \(A_{i}=(0, i),\) that is, the set of real numbers \(x\) with \(0< x< i .\) d) \(A_{i}=(i, \infty),\) that is, the set of real numbers \(x\) with \(x>i .\)

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