명제와 조건

Assume the universal set for all variables is $\mathbb{Z}$ and let $P( x )$ be the predicate “$x^2 \leq 4$.”

(i) Find two values of $x$ for which $P( x )$ is false.

(ii) Find two values of $x$ for which $P( x )$ is true.

(iii) Use the roster method to specify the set of all $x$ for which $P( x )$ is true.

Assume the universal set for all variables is $\mathbb{Z}$, and let $R( {x,\;y,\;z} )$ be the predicate “$x^2 + y^2 = z^2$.”

(i) Find two different examples for which $R( {x, y, z})$ is false.

(ii) Find two different examples for which $R( {x, y, z})$ is true.

Let $P( x )$ be the predicate “$x^2 \leq 9$.”

If the universal set is $\mathbb{R}$, describe the truth set of $P(x)$ using English and write the truth set of $P( x )$ using set builder notation.

Let $P( x )$ be the predicate “$x^2 \leq 9$.”

If the universal set is $\mathbb{Z}$, then what is the truth set of $P( x )$? Describe this set using English and then use the roster method to specify all the elements of this truth set.

Use the roster method to specify the truth set for each of the following open sentences. The universal set for each open sentence is the set of integers $\mathbb{Z}$.

$n + 7 = 4$.

Use the roster method to specify the truth set for each of the following open sentences. The universal set for each open sentence is the set of integers $\mathbb{Z}$.

$n^2 = 64$.

Use the roster method to specify the truth set for each of the following open sentences. The universal set for each open sentence is the set of integers $\mathbb{Z}$.

$\sqrt{n} \in \mathbb{N}$ and $n$ is less than 50.

Use the roster method to specify the truth set for each of the following open sentences. The universal set for each open sentence is the set of integers $\mathbb{Z}$.

$n$ is an odd integer that is greater than 2 and less than 14.

Use the roster method to specify the truth set for each of the following open sentences. The universal set for each open sentence is the set of integers $\mathbb{Z}$.

$n$ is an even integer that is greater than 10.

For each of the following statements

• (1) Write the statement in the form of an English sentence that does not use the symbols for quantifiers. - (2) Write the negation of the statement in a symbolic form that does not use the negation symbol. - (3) Write the negation of the statement in the form of an English sentence that does not use the symbols for quantifiers.

$\left( \forall a \in \mathbb{R}\right) \left( a + 0 = a \right)$.

For each of the following statements

• (1) Write the statement in the form of an English sentence that does not use the symbols for quantifiers. - (2) Write the negation of the statement in a symbolic form that does not use the negation symbol. - (3) Write the negation of the statement in the form of an English sentence that does not use the symbols for quantifiers.

$\left( \forall x \in \mathbb{R} \right) \left[ \sin ( {2x} ) = 2 ( {\sin x} ) ( {\cos x} ) \right]$.

For each of the following statements

• (1) Write the statement in the form of an English sentence that does not use the symbols for quantifiers. - (2) Write the negation of the statement in a symbolic form that does not use the negation symbol. - (3) Write the negation of the statement in the form of an English sentence that does not use the symbols for quantifiers.

$\left( \forall x \in \mathbb{R} \right) \left( \tan ^2 x + 1 = \sec ^2 x \right)$.

For each of the following statements

• (1) Write the statement in the form of an English sentence that does not use the symbols for quantifiers. - (2) Write the negation of the statement in a symbolic form that does not use the negation symbol. - (3) Write the negation of the statement in the form of an English sentence that does not use the symbols for quantifiers.

$\left( \exists x \in \mathbb{Q} \right) \left( x^2 - 3x - 7 = 0 \right)$.

For each of the following statements

• (1) Write the statement in the form of an English sentence that does not use the symbols for quantifiers. - (2) Write the negation of the statement in a symbolic form that does not use the negation symbol. - (3) Write the negation of the statement in the form of an English sentence that does not use the symbols for quantifiers.

$\left( \exists x \in \mathbb{R} \right) \left( x^2 + 1 = 0 \right)$.

Use counterexamples to explain why each of the following statements is false.

For each integer $n$, $\left( n^2 + n + 1 \right)$ is a prime number.

Use counterexamples to explain why each of the following statements is false.

For each real number $x$, if $x$ is positive, then $2x^2 > x$.

An integer $n$ is a multiple of 3 provided that there exists an integer $k$ such that $n = 3k$.

Write this definition in symbolic form using quantifiers by completing the following:

An integer

n

is a multiple of 3 provided that

.

An integer $n$ is a multiple of 3 provided that there exists an integer $k$ such that $n = 3k$.

Give several examples of integers (including negative integers) that are multiples of 3.

An integer $n$ is a multiple of 3 provided that there exists an integer $k$ such that $n = 3k$.

Give several examples of integers (including negative integers) that are not multiples of 3.

An integer $n$ is a multiple of 3 provided that there exists an integer $k$ such that $n = 3k$.

Use the symbolic form of the definition of a multiple of 3 to complete the following sentence: “An integer $n$ is not a multiple of 3 provided that ….”