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Chapter 3: Problem 12

Let \(p(x)\) stand for " \(x\) is a prime," \(q(x)\) for " \(x\) is even," and \(r(x, y)\) stand for " \(x=y . "\) Use these three symbolic statements and appropriate logical notation to write the statement "There is one and only one even prime." (Use the set \(Z^{+}\)of positive integers for your universe.)

Short Answer

Expert verified
\( \exists x ((p(x) \land q(x)) \land \forall y ((p(y) \land q(y)) \rightarrow r(y,x))) \)

Step by step solution

01

Understanding the Statement

We need to express the statement "There is one and only one even prime" using logical notation and the predicates given: 1) \( p(x) \): \( x \) is a prime, 2) \( q(x) \): \( x \) is even, and 3) \( r(x, y) \): \( x = y \). Our universe of discourse here is the set of positive integers, \( \mathbb{Z}^+ \).
02

Defining the Conditions

We need to reflect two conditions in our notation: a) An integer is a prime and even, and b) Only one such integer exists. This calls for using logical quantifiers and combining the three predicates. Firstly, identify such an integer \( x \) and ensure \( p(x) \) and \( q(x) \) holds, meaning \( x \) is both prime and even.
03

Writing the Condition for One Specific Value

Our logical expression should say: "There exists an \( x \) such that \( x \) is even and prime." This can be represented as \( \exists x (p(x) \land q(x)) \).
04

Ensuring Uniqueness

Beyond existence, it must be the only such integer. This can be expressed as: "For all integers \( y \), if \( y \) is even and prime, then \( y = x \)," which can be presented as: \( \forall y ((p(y) \land q(y)) \rightarrow r(y,x)) \).
05

Combining Both Conditions

To combine the existence of such an integer with the statement that it is the only one, we use both in a single statement: \( \exists x ((p(x) \land q(x)) \land \forall y ((p(y) \land q(y)) \rightarrow r(y,x))) \). This states there is one specific integer that is even and prime, and if any integer satisfies those predicates, it is identical to that specific value.

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Key Concepts

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

Logical Quantifiers
Logical quantifiers are symbols used in predicate logic to express the frequency or number of elements that satisfy a given property within a specific domain. Understanding these symbols is crucial when translating real-world statements into formal logical expressions. Two primary types of logical quantifiers are:
  • Universal Quantifier (\( \forall \)): This symbol is read as "for all" and indicates that a certain property or condition holds true for every element in a given set. In other words, \( \forall x P(x) \) means the property \( P \) is true for all values of \( x \).
  • Existential Quantifier (\( \exists \)): This quantifier indicates that there is at least one element in the set for which the property holds true. It is read as "there exists." For example, \( \exists x P(x) \) asserts that there is some \( x \) for which \( P(x) \) is true.
These quantifiers can be combined to form complex statements that involve existence as well as uniqueness, like in the exercise statement "There is one and only one even prime." Here, we not only need to find an existing case (existential) but also ensure it's the only one (uniqueness). This requires a thoughtful application of logic, combining both existential and universal quantifiers, as shown in the solution.
Prime Numbers
Prime numbers are the building blocks of arithmetic in mathematics and play a crucial role in number theory. A prime number is defined as a natural number greater than 1 that has no divisors other than 1 and itself. In simpler terms, a prime number can only be divided evenly by 1 and the number itself. For example:
  • 2 - This is the smallest and the only even prime number. It is divisible only by 1 and 2.
  • 3 - Another prime, which can only be divided without remainder by 1 and 3.
  • 5, 7, 11, 13 - These numbers follow the same rule, being divisible only by themselves and unity.
Understanding prime numbers is crucial for translating the exercise statement into logical terms. Recognizing that there is only one even prime number, which is 2, helps us intuitively understand why the logical statement asserting this uniqueness is valid. The use of prime numbers in logical statements showcases their role not only within mathematics but also in logical reasoning and computer science.
Even Numbers
Even numbers are integers that can be exactly divided by 2 鈥� in other words, they have 2 as one of their divisors. This is an essential concept in determining properties of numbers and is straightforward:
  • Examples of even numbers include 2, 4, 6, 8, 10, etc. These all leave no remainder when divided by 2.
  • In contrast, numbers like 1, 3, and 5 are not even, as they would have a remainder when divided by 2.
Even numbers are significant in the context of this exercise because we need to pinpoint a specific characteristic 鈥� an even number that is also prime. This highlights the number 2 uniquely, as it is the only even number that does not share factors with any other number beyond itself and 1. Thus, for the logical exercise, understanding even numbers extends beyond identification and into recognizing their singular intersection with prime properties.

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Most popular questions from this chapter

Why is \((\exists x \in U(p(x))) \wedge(\exists y \in U(q(y)))\) not equivalent to \(\exists z \in U(p(z) \wedge q(z)) ?\) Are the statements \((\exists x \in U(p(x))) \vee(\exists y\) \(\in U(q(y)))\) and \(\exists z \in U(p(z) \vee q(z))\) equivalent?

Using \(\mathrm{T}\) to stand for a statement that is always true and \(\mathrm{F}\) to stand for a statement that is always false, give a simplified form of each of the following statements. a. \(\mathrm{T} \wedge \mathrm{s}\) b. \(F \wedge s\) c. \(\mathrm{T} \vee \mathrm{s}\) d. \(\mathrm{F} \vee \mathrm{s}\)

Rewrite the following statement without any negations: "There is no positive integer \(n\) such that for all integers \(m>n\), all polynomial equations \(p(x)=0\) of degree \(m\) have no real numbers for solutions."

Show that the statements \(s \Rightarrow t\) and \(\neg s \vee t\) are equivalent.

Write the definition of a greatest common divisor of \(m\) and \(n\) in such a way that all quantifiers are explicit and expressed explicitly as "for all" or "there exists." Write the part of Euclid's extended greatest common divisor theorem (Theorem 2.14) that relates the greatest common divisor of \(m\) and \(n\) algebraically to \(m\) and \(n\). Again, make sure all quantifiers are explicit and expressed explicitly as "for all" or "there exists."
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