명제와 조건

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.

$\left( {\exists x \in \mathbb{Q}} \right)\left( {\sqrt 2 \lt x \lt \sqrt 3 } \right)$.

Note: The sentence “$\sqrt 2 \lt x \lt \sqrt 3$” is actually a conjunction. It means $\sqrt 2 \lt x$ and $x \lt \sqrt 3$.

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.

$\left( {\forall x \in \mathbb{Z}} \right)\left( {\text{ If } x^2 \text{ is odd, then } x\text{ is odd } } \right)$.

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.

$\left( {\forall n \in \mathbb{N}} \right)$ [If $n$ is a perfect square, then $\left( {2^n - 1} \right)$ is not a prime number].

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 $\ldots$.