집합과 명제 전체 문제

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?

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

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?

$F\left( x \right) = - x^4 + x^3 + 9$

Following is a statement of a theorem which can be proven using the quadratic formula. 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 $ac \lt 0$, then the function $f$ has two $x$-intercepts.

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 the quadratic formula. 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 $ac \lt 0$, then the function $f$ has two $x$-intercepts.

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 the quadratic formula. 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 $ac \lt 0$, then the function $f$ has two $x$-intercepts.

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 the quadratic formula. 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 $ac \lt 0$, then the function $f$ has two $x$-intercepts.

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 the quadratic formula. 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 $ac \lt 0$, then the function $f$ has two $x$-intercepts.

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

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

Following is a statement of a theorem which can be proven using the quadratic formula. 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 $ac \lt 0$, then the function $f$ has two $x$-intercepts.

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

$F\left( x \right) = - x^4 + x^3 + 9$

Following is a statement of a theorem about certain cubic equations. For this theorem, $b$ represents a real number.

• Theorem A If $f$ is a cubic function of the form $f(x) = x^3 - x + b$ and $b > 1$, then the function $f$ has exactly one $x$-intercept.

Following is another theorem about $x$-intercepts of functions:

• Theorem B If $f$ and $g$ are functions with $g(x) = k \cdot f(x)$, where $k$ is a nonzero real number, then $f$ and $g$ have exactly the same $x$-intercepts.

Using only these two theorems and some simple algebraic manipulations, what can be concluded about the functions given by the following formulas?

$f(x) = x^3 - x + 7$

Following is a statement of a theorem about certain cubic equations. For this theorem, $b$ represents a real number.

• Theorem A If $f$ is a cubic function of the form $f(x) = x^3 - x + b$ and $b > 1$, then the function $f$ has exactly one $x$-intercept.

Following is another theorem about $x$-intercepts of functions:

• Theorem B If $f$ and $g$ are functions with $g(x) = k \cdot f(x)$, where $k$ is a nonzero real number, then $f$ and $g$ have exactly the same $x$-intercepts.

Using only these two theorems and some simple algebraic manipulations, what can be concluded about the functions given by the following formulas?

$g(x) = x^3 + x + 7$

Following is a statement of a theorem about certain cubic equations. For this theorem, $b$ represents a real number.

• Theorem A If $f$ is a cubic function of the form $f(x) = x^3 - x + b$ and $b > 1$, then the function $f$ has exactly one $x$-intercept.

Following is another theorem about $x$-intercepts of functions:

• Theorem B If $f$ and $g$ are functions with $g(x) = k \cdot f(x)$, where $k$ is a nonzero real number, then $f$ and $g$ have exactly the same $x$-intercepts.

Using only these two theorems and some simple algebraic manipulations, what can be concluded about the functions given by the following formulas?

$h(x) = -x^3 + x - 5$

Following is a statement of a theorem about certain cubic equations. For this theorem, $b$ represents a real number.

• Theorem A If $f$ is a cubic function of the form $f(x) = x^3 - x + b$ and $b > 1$, then the function $f$ has exactly one $x$-intercept.

Following is another theorem about $x$-intercepts of functions:

• Theorem B If $f$ and $g$ are functions with $g(x) = k \cdot f(x)$, where $k$ is a nonzero real number, then $f$ and $g$ have exactly the same $x$-intercepts.

Using only these two theorems and some simple algebraic manipulations, what can be concluded about the functions given by the following formulas?

$k(x) = 2x^3 + 2x + 3$

Following is a statement of a theorem about certain cubic equations. For this theorem, $b$ represents a real number.

• Theorem A If $f$ is a cubic function of the form $f(x) = x^3 - x + b$ and $b > 1$, then the function $f$ has exactly one $x$-intercept.

Following is another theorem about $x$-intercepts of functions:

• Theorem B If $f$ and $g$ are functions with $g(x) = k \cdot f(x)$, where $k$ is a nonzero real number, then $f$ and $g$ have exactly the same $x$-intercepts.

Using only these two theorems and some simple algebraic manipulations, what can be concluded about the functions given by the following formulas?

$r(x) = x^4 - x + 11$

Following is a statement of a theorem about certain cubic equations. For this theorem, $b$ represents a real number.

• Theorem A If $f$ is a cubic function of the form $f(x) = x^3 - x + b$ and $b > 1$, then the function $f$ has exactly one $x$-intercept.

Following is another theorem about $x$-intercepts of functions:

• Theorem B If $f$ and $g$ are functions with $g(x) = k \cdot f(x)$, where $k$ is a nonzero real number, then $f$ and $g$ have exactly the same $x$-intercepts.

Using only these two theorems and some simple algebraic manipulations, what can be concluded about the functions given by the following formulas?

$F(x) = 2x^3 - 2x + 7$

Using what you learned in this section, answer the following.

Is the set of natural numbers closed under division?