명제와 조건

Which of the following sentences are statements? Do not worry about determining the truth value of those that are statements; just determine whether each sentence is a statement or not.

$2 \cdot 7 + 8 = 22$.

Which of the following sentences are statements? Do not worry about determining the truth value of those that are statements; just determine whether each sentence is a statement or not.

$2x + 5y = 7$.

Which of the following sentences are statements? Do not worry about determining the truth value of those that are statements; just determine whether each sentence is a statement or not.

There are integers $x$ and $y$ such that $2x + 5y = 7$.

Which of the following sentences are statements? Do not worry about determining the truth value of those that are statements; just determine whether each sentence is a statement or not.

Given a line $L$ and a point $P$ not on that line and in the same plane, there is a unique line in that plane through $P$ that does not intersect $L$.

Which of the following sentences are statements? Do not worry about determining the truth value of those that are statements; just determine whether each sentence is a statement or not.

$\left( {a + b} \right)^3 = a^3 + 3a^2b + 3ab^2 + b^3$.

Which of the following sentences are statements? Do not worry about determining the truth value of those that are statements; just determine whether each sentence is a statement or not.

For all real numbers $a$ and $b$, $\left( {a + b} \right)^3 = a^3 + 3a^2b + 3ab^2 + b^3$.

Which of the following sentences are statements? Do not worry about determining the truth value of those that are statements; just determine whether each sentence is a statement or not.

Does the equation $3x^2 - 5x - 7 = 0$ have two real number solutions?

Which of the following sentences are statements? Do not worry about determining the truth value of those that are statements; just determine whether each sentence is a statement or not.

If $ABC$ is a right triangle with right angle at vertex $B$, and if $D$ is the midpoint of the hypotenuse, then the line segment connecting vertex $B$ to $D$ is half the length of the hypotenuse.

Which of the following sentences are statements? Do not worry about determining the truth value of those that are statements; just determine whether each sentence is a statement or not.

There do not exist three integers $x$, $y$, and $z$ such that $x^3 + y^3 = z^3$.

“If $n$ is a positive integer, then $(n^2-n+41)$ is a prime number.” (Remember that a prime number is a positive integer greater than 1 whose only positive factors are 1 and itself.) To explore whether or not this statement is true, try using (and recording your results) for $n=1$, $n=2$, $n=3$, $n=4$, $n=5$, and $n=10$. Then record the results for at least four other values of $n$. Does this conditional statement appear to be true?

Answer each of the following questions.

Is the set of rational numbers closed under addition? Explain.

Answer each of the following questions.

Is the set of integers closed under division? Explain.

Answer each of the following questions.

Is the set of rational numbers closed under subtraction? Explain.

Which of the following sentences are statements?

$3^2 + 4^2 = 5^2$.

Which of the following sentences are statements?

$a^2 + b^2 = c^2$.

Which of the following sentences are statements?

There exist integers $a$, $b$, and $c$ such that $a^2 = b^2 + c^2$.

Which of the following sentences are statements?

If $x^2 = 4$, then $x = 2$.

Which of the following sentences are statements?

For each real number $x$, if $x^2 = 4$, then $x = 2$.

Which of the following sentences are statements?

For each real number $t$, $\sin ^2 t + \cos ^2 t = 1$.

Which of the following sentences are statements?

$\sin x \lt \sin \! \left( {\dfrac{\pi }{4}} \right)$.