집합과 명제 전체 문제

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?

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

Which of the following sentences are statements?

If $n$ is a prime number, then $n^2$ has three positive factors.

Which of the following sentences are statements?

$1 + \tan ^2 \theta = \sec ^2 \theta$.

Which of the following sentences are statements?

Every rectangle is a parallelogram.

Which of the following sentences are statements?

Every even natural number greater than or equal to 4 is the sum of two prime numbers.

Identify the hypothesis and the conclusion for each of the following conditional statements.

If $n$ is a prime number, then $n^2$ has three positive factors.

Identify the hypothesis and the conclusion for each of the following conditional statements.

If $a$ is an irrational number and $b$ is an irrational number, then $a \cdot b$ is an irrational number.

Identify the hypothesis and the conclusion for each of the following conditional statements.

If $p$ is a prime number, then $p = 2$ or $p$ is an odd number.

Identify the hypothesis and the conclusion for each of the following conditional statements.

If $p$ is a prime number and $p \ne 2$, then $p$ is an odd number.

Identify the hypothesis and the conclusion for each of the following conditional statements.

If $p \ne 2$ and $p$ is an even number, then $p$ is not prime.

Determine the conditions under which each of the following conditional sentences will be a true statement.

If $a+2=5$, then $8\lt 5$.

Determine the conditions under which each of the following conditional sentences will be a true statement.

If $5\lt 8$, then $a+2=5$.

Let $P$ be the statement “Student X passed every assignment in Calculus I,” and let $Q$ be the statement “Student X received a grade of C or better in Calculus I.”

What does it mean for $P$ to be true? What does it mean for $Q$ to be true?

Let $P$ be the statement “Student X passed every assignment in Calculus I,” and let $Q$ be the statement “Student X received a grade of C or better in Calculus I.”

Suppose that Student X passed every assignment in Calculus I and received a grade of B$-$, and that the instructor made the statement $P \to Q$. Would you say that the instructor lied or told the truth?

Let $P$ be the statement “Student X passed every assignment in Calculus I,” and let $Q$ be the statement “Student X received a grade of C or better in Calculus I.”

Suppose that Student X passed every assignment in Calculus I and received a grade of C$-$, and that the instructor made the statement $P \to Q$. Would you say that the instructor lied or told the truth?