집합과 명제 전체 문제

For each of the following statements

• Write the statement as an English sentence that does not use the symbols for quantifiers. - Write the negation of the statement in symbolic form in which the negation symbol is not used. - Write a useful negation of the statement in an English sentence that does not use the symbols for quantifiers.

$\left( {\forall n \in \mathbb{N}} \right)\left( {n^2 - n + 41} \text{ is a prime number } \right)$.

For each of the following statements

• Write the statement as an English sentence that does not use the symbols for quantifiers. - Write the negation of the statement in symbolic form in which the negation symbol is not used. - Write a useful negation of the statement in an English sentence that does not use the symbols for quantifiers.

$\left( {\exists x \in \mathbb{R}} \right) \left( {\cos ( {2x} ) = 2( {\cos x} )} \right)$.

Write each of the following statements as an English sentence that does not use the symbols for quantifiers.

$\left( {\exists m} \in \mathbb{Z} \right)\left( {\exists n} \in \mathbb{Z} \right)\left( {m > n} \right)$

Write each of the following statements as an English sentence that does not use the symbols for quantifiers.

$\left( {\exists m} \in \mathbb{Z} \right)\left( {\forall n} \in \mathbb{Z} \right)\left( {m > n} \right)$

Write each of the following statements as an English sentence that does not use the symbols for quantifiers.

$\left( {\forall m} \in \mathbb{Z} \right)\left( {\exists n} \in \mathbb{Z} \right)\left( {m > n} \right)$

Write each of the following statements as an English sentence that does not use the symbols for quantifiers.

$\left( {\forall m}\in \mathbb{Z} \right)\left( {\forall n} \in \mathbb{Z} \right)\left( {m > n} \right)$

Write each of the following statements as an English sentence that does not use the symbols for quantifiers.

$\left( {\exists n} \in \mathbb{Z} \right)\left( {\forall m} \in \mathbb{Z} \right)\left( {m^2 > n} \right)$

Write each of the following statements as an English sentence that does not use the symbols for quantifiers.

$\left( {\forall n} \in \mathbb{Z} \right)\left( {\exists m} \in \mathbb{Z} \right)\left( {m^2 > n} \right)$

Assume that the universal set is $\mathbb{Z}$. Consider the following sentence:

$$\left( {\exists t \in \mathbb{Z}} \right)\left( {t \cdot x = 20} \right)$$

.

.

Explain why this sentence is an open sentence and not a statement.

Assume that the universal set is $\mathbb{Z}$. Consider the following sentence:

$$\left( {\exists t \in \mathbb{Z}} \right)\left( {t \cdot x = 20} \right)$$

.

.

What is the truth set of the open sentence $\left( {\exists t \in \mathbb{Z}} \right)\left( {t \cdot x = 20} \right)$?

Assume that the universal set is $\mathbb{R}$. Consider the following sentence:

$$\left( {\exists t \in \mathbb{R}} \right)\left( {t \cdot x = 20} \right)$$

.

Explain why this sentence is an open sentence and not a statement.

Assume that the universal set is $\mathbb{R}$. Consider the following sentence:

$$\left( {\exists t \in \mathbb{R}} \right)\left( {t \cdot x = 20} \right)$$

.

What is the truth set of the open sentence $\left( {\exists t \in \mathbb{R}} \right)\left( {t \cdot x = 20} \right)$?

Let $\mathbb{Z}^*$ be the set of all nonzero integers.

Use a counterexample to explain why the following statement is false:

For each

x \in \Z^*

, there exists a

y \in \Z^*

such that

xy = 1

.

An integer $m$ is said to have the divides property

provided that for all integers $a$ and $b$, if $m$ divides $ab$, then $m$ divides $a$ or $m$ divides $b$.

Using the symbols for quantifiers, write what it means to say that the integer $m$ has the divides property.

An integer $m$ is said to have the divides property

provided that for all integers $a$ and $b$, if $m$ divides $ab$, then $m$ divides $a$ or $m$ divides $b$.

Using the symbols for quantifiers, write what it means to say that the integer $m$ does not have the divides property.

An integer $m$ is said to have the divides property

provided that for all integers $a$ and $b$, if $m$ divides $ab$, then $m$ divides $a$ or $m$ divides $b$.

Write an English sentence stating what it means to say that the integer $m$ does not have the divides property.

In calculus, we define a function $f$ with domain $\mathbb{R}$ to be strictly increasing

provided that for all real numbers $x$ and $y$, $f\left( x \right) \lt f\left( y \right)$ whenever $x \lt y$. Complete each of the following sentences using the appropriate symbols for quantifiers:

A function $f$ with domain $\mathbb{R}$ is strictly increasing provided that $\ldots$.

In calculus, we define a function $f$ with domain $\mathbb{R}$ to be strictly increasing

provided that for all real numbers $x$ and $y$, $f\left( x \right) \lt f\left( y \right)$ whenever $x \lt y$. Complete each of the following sentences using the appropriate symbols for quantifiers:

A function $f$ with domain $\mathbb{R}$ is not strictly increasing provided that $\ldots$.

In calculus, we define a function $f$ with domain $\mathbb{R}$ to be strictly increasing

provided that for all real numbers $x$ and $y$, $f\left( x \right) \lt f\left( y \right)$ whenever $x \lt y$. Complete each of the following sentences using the appropriate symbols for quantifiers:

Complete the following sentence in English without using symbols for quantifiers:

A function $f$ with domain $\mathbb{R}$ is not strictly increasing provided that $\ldots$.

The following exercises contain definitions or results from more advanced mathematics courses. Even though we may not understand all of the terms involved, it is still possible to recognize the structure of the given statements and write a meaningful negation of that statement.

In abstract algebra, an operation $*$ on a set $A$ is called a commutative operation

provided that for all $x, y \in A$, $x * y = y * x$. Carefully explain what it means to say that an operation $*$ on a set $A$ is not a commutative operation.