명제와 조건

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?

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.”

Now suppose that Student X did not pass two assignments in Calculus I and received a grade of D, and that the instructor made the statement $P \to Q$. Would you say that the instructor lied or told the truth?

Following is a statement of a theorem which can be proven using calculus or precalculus mathematics. For this theorem, $a$, $b$, and $c$ are real numbers.

• Theorem If $f$ is a quadratic function of the form $f\left( x \right) = ax^2 + bx + c$ and $a \lt 0$, then the function $f$ has a maximum value when $x = \dfrac{{ - b}}{{2a}}$.

Using only this theorem, what can be concluded about the functions given by the following formulas?

$g\left( x \right) = - 8x^2 + 5x - 2$

Following is a statement of a theorem which can be proven using calculus or precalculus mathematics. For this theorem, $a$, $b$, and $c$ are real numbers.

• Theorem If $f$ is a quadratic function of the form $f\left( x \right) = ax^2 + bx + c$ and $a \lt 0$, then the function $f$ has a maximum value when $x = \dfrac{{ - b}}{{2a}}$.

Using only this theorem, what can be concluded about the functions given by the following formulas?

$h\left( x \right) = - \dfrac{1}{3}x^2 + 3x$

Following is a statement of a theorem which can be proven using calculus or precalculus mathematics. For this theorem, $a$, $b$, and $c$ are real numbers.

• Theorem If $f$ is a quadratic function of the form $f\left( x \right) = ax^2 + bx + c$ and $a \lt 0$, then the function $f$ has a maximum value when $x = \dfrac{{ - b}}{{2a}}$.

Using only this theorem, what can be concluded about the functions given by the following formulas?

$k\left( x \right) = 8x^2 - 5x - 7$

Following is a statement of a theorem which can be proven using calculus or precalculus mathematics. For this theorem, $a$, $b$, and $c$ are real numbers.

• Theorem If $f$ is a quadratic function of the form $f\left( x \right) = ax^2 + bx + c$ and $a \lt 0$, then the function $f$ has a maximum value when $x = \dfrac{{ - b}}{{2a}}$.

Using only this theorem, what can be concluded about the functions given by the following formulas?

$j\left( x \right) = - \dfrac{{71}}{{99}}x^2 + 210$

Following is a statement of a theorem which can be proven using calculus or precalculus mathematics. For this theorem, $a$, $b$, and $c$ are real numbers.

• Theorem If $f$ is a quadratic function of the form $f\left( x \right) = ax^2 + bx + c$ and $a \lt 0$, then the function $f$ has a maximum value when $x = \dfrac{{ - b}}{{2a}}$.

Using only this theorem, what can be concluded about the functions given by the following formulas?

$f\left( x \right) = - 4x^2 - 3x + 7$