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Chapter 2: Problem 4

Write each of the following statements as an English sentence that does not use the symbols for quantifiers. * (a) \((\exists m \in \mathbb{Z})(\exists n \in \mathbb{Z})(m>n)\) (d) \((\forall m \in \mathbb{Z})(\forall n \in \mathbb{Z})(m>n)\) (b) \((\exists m \in \mathbb{Z})(\forall n \in \mathbb{Z})(m>n)\) (e) \((\exists n \in \mathbb{Z})(\forall m \in \mathbb{Z})\left(m^{2}>n\right)\) (c) \((\forall m \in \mathbb{Z})(\exists n \in \mathbb{Z})(m>n)\) (f) \((\forall n \in \mathbb{Z})(\exists m \in \mathbb{Z})\left(m^{2}>n\right)\)

Short Answer

Expert verified
In summary, the English translations of the given statements are: (a) "There exist integers m and n such that m is greater than n." (b) "There exists an integer m such that for all integers n, m is greater than n." (c) "For every integer m, there exists an integer n such that m is greater than n." (d) "For all integers m and n, m is greater than n." (Note: This statement is not true.) (e) "There exists an integer n such that for all integers m, the square of m is greater than n." (f) "For every integer n, there exists an integer m such that the square of m is greater than n."

Step by step solution

01

Statement (a)

For statement (a) \((\exists m \in \mathbb{Z})(\exists n \in \mathbb{Z})(m>n)\), it can be translated as "there exist integers m and n such that m is greater than n."
02

Statement (b)

For statement (b) \((\exists m \in \mathbb{Z})(\forall n \in \mathbb{Z})(m>n)\), it can be translated as "there exists an integer m such that for all integers n, m is greater than n."
03

Statement (c)

For statement (c) \((\forall m \in \mathbb{Z})(\exists n \in \mathbb{Z})(m>n)\), it can be translated as "for every integer m, there exists an integer n such that m is greater than n."
04

Statement (d)

For statement (d) \((\forall m \in \mathbb{Z})(\forall n \in \mathbb{Z})(m>n)\), it can be translated as "for all integers m and n, m is greater than n." However, this statement is not true, since it is not possible for all integers m to be greater than all integers n.
05

Statement (e)

For statement (e) \((\exists n \in \mathbb{Z})(\forall m \in \mathbb{Z})\left(m^{2}>n\right)\), it can be translated as "there exists an integer n such that for all integers m, the square of m is greater than n."
06

Statement (f)

For statement (f) \((\forall n \in \mathbb{Z})(\exists m \in \mathbb{Z})\left(m^{2}>n\right)\), it can be translated as "for every integer n, there exists an integer m such that the square of m is greater than n."

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Key Concepts

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

Translation of Symbolic Logic
Understanding symbolic logic is like decoding a secret language. It transforms complex ideas into simple symbols. This often helps in mathematics and computer science. But turning these symbols back into plain English is critical. You make sure the original meanings remain intact.

Let's explore a simple example. Think of the expression \((\exists m \in \mathbb{Z})(\exists n \in \mathbb{Z})(m>n)\). In translation, it reads as: "there exist integers \(m\) and \(n\) such that \(m\) is greater than \(n\)."

What this does is make complex statements easier to analyze. However, if you don鈥檛 fully grasp how translation works, symbolic logic can seem intimidating. Here's a simple way to break down these translations:
  • If you see \( \exists \), think "there exists." It means at least one instance make this true.
  • With \( \forall \), imagine "for all" or "every." This represents a universal application.

By mastering these translations, one can approach problems with a clearer mind.
Integers in Mathematical Logic
In the realm of mathematical logic, integers (\(\mathbb{Z}\)) are the bedrock of many logical statements. They include positive and negative numbers and zero. In logic, they are used to present arguments without any fractions or decimals.

Consider this scenario with integers: \((\forall m \in \mathbb{Z})(\exists n \in \mathbb{Z})(m>n)\). This means "for every integer \(m\), there exists an integer \(n\) such that \(m\) is greater than \(n\)."
Using integers this way simplifies analysis as they maintain discrete values. They are predictable and follow basic properties:
  • Addition and subtraction always result in another integer.
  • Multiplication likewise results in an integer.

Understanding their role helps you construct and deconstruct logical statements efficiently.
Universal and Existential Quantifiers
Quantifiers are powerful symbols in logic, making broad statements more precise and meaningful. The two main types are universal and existential quantifiers, represented as \(\forall\) and \(\exists\) respectively.

  • The universal quantifier \(\forall\) signifies "for all" or "for every," meaning the statement must be universally true.
  • The existential quantifier \(\exists\) indicates "there exists" or "there is," signifying that there is at least one instance where the statement is true.

Take for example, \((\exists m \in \mathbb{Z})(\forall n \in \mathbb{Z})(m>n)\). This means "there exists an integer \(m\) such that for all integers \(n\), \(m\) is greater than \(n\)."
Recognizing and using these quantifiers equips you to formulate or interpret vast ideas precisely. They are crucial in mathematics for defining properties, setting conditions, or even proving theories.

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Most popular questions from this chapter

Suppose that \(P\) and \(Q\) are statements for which \(P \rightarrow Q\) is false. What conclusion (if any) can be made about the truth value of each of the following statements? (a) \(\neg P \rightarrow Q\) (b) \(Q \rightarrow P\) (c) \(P \vee Q\)

Suppose each of the following statements is true. \- Laura is in the seventh grade. \- Laura got an \(\mathrm{A}\) on the mathematics test or Sarah got an \(\mathrm{A}\) on the mathematics test. \- If Sarah got an \(\mathrm{A}\) on the mathematics test, then Laura is not in the seventh grade. If possible, determine the truth value of each of the following statements. Carefully explain your reasoning. (a) Laura got an \(\mathrm{A}\) on the mathematics test. (b) Sarah got an \(\mathrm{A}\) on the mathematics test. (c) Either Laura or Sarah did not get an \(\mathrm{A}\) on the mathematics test.

Suppose we are trying to prove the following for integers \(x\) and \(y\) : If \(x \cdot y\) is even, then \(x\) is even or \(y\) is even. We notice that we can write this statement in the following symbolic form: $$ P \rightarrow(Q \vee R) $$ where \(P\) is " \(x \cdot y\) is even," \(Q\) is " \(x\) is even," and \(R\) is " \(y\) is even." (a) Write the symbolic form of the contrapositive of \(P \rightarrow(Q \vee R)\). Then use one of De Morgan's Laws (Theorem 2.5 ) to rewrite the hypothesis of this conditional statement. (b) Use the result from Part (13a) to explain why the given statement is logically equivalent to the following statement: If \(x\) is odd and \(y\) is odd, then \(x \cdot y\) is odd. The two statements in this activity are logically equivalent. We now have the choice of proving either of these statements. If we prove one, we prove the other, or if we show one is false, the other is also false. The second statement is Theorem \(1.8,\) which was proven in Section 1.2 .

In calculus, we define a function \(f\) to be continuous at a real number \(a\) provided that for every \(\varepsilon>0,\) there exists a \(\delta>0\) such that if \(|x-a|<\delta\), then \(|f(x)-f(a)|<\varepsilon\). Note: The symbol \(\varepsilon\) is the lowercase Greek letter epsilon, and the symbol \(\delta\) is the lowercase Greek letter delta. Complete each of the following sentences using the appropriate symbols for quantifiers: (a) A function \(f\) is continuous at the real number a provided that .... (b) A function \(f\) is not continuous at the real number \(a\) provided that \(\ldots .\) Complete the following sentence in English without using symbols for quantifiers: (c) A function \(f\) is not continuous at the real number \(a\) provided that \(\ldots .\)

Let \(a, b,\) and \(c\) be integers. Consider the following conditional statement: If \(a\) divides \(b c,\) then \(a\) divides \(b\) or \(a\) divides \(c\). Which of the following statements have the same meaning as this conditional statement and which ones are negations of this conditional statement? The note for Exercise (10) also applies to this exercise. (a) If \(a\) divides \(b\) or \(a\) divides \(c,\) then \(a\) divides \(b c\). (b) If \(a\) does not divide \(b\) or \(a\) does not divide \(c,\) then \(a\) does not divide \(b c\). (c) \(a\) divides \(b c, a\) does not divide \(b\), and \(a\) does not divide \(c\). (d) If \(a\) does not divide \(b\) and \(a\) does not divide \(c,\) then \(a\) does not divide \(b c\) (e) \(a\) does not divide \(b c\) or \(a\) divides \(b\) or \(a\) divides \(c\). (f) If \(a\) divides \(b c\) and \(a\) does not divide \(c,\) then \(a\) divides \(b\). (g) If \(a\) divides \(b c\) or \(a\) does not divide \(b\), then \(a\) divides \(c\).
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