집합과 명제 전체 문제

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?

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

Is the set of rational numbers closed under division?

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

Is the set of nonzero rational numbers closed under division?

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

Is the set of positive rational numbers closed under division?

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

Is the set of positive real numbers closed under subtraction?

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

Is the set of negative rational numbers closed under division?

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

Is the set of negative integers closed under addition?

Recall that a quadrilateral is a four-sided polygon. Let $S$ represent the following true conditional statement:

If a quadrilateral is a square, then it is a rectangle.

Write this conditional statement in English using

the word “whenever”

Recall that a quadrilateral is a four-sided polygon. Let $S$ represent the following true conditional statement:

If a quadrilateral is a square, then it is a rectangle.

Write this conditional statement in English using

the phrase “only if”

Recall that a quadrilateral is a four-sided polygon. Let $S$ represent the following true conditional statement:

If a quadrilateral is a square, then it is a rectangle.

Write this conditional statement in English using

the phrase “is necessary for”

Recall that a quadrilateral is a four-sided polygon. Let $S$ represent the following true conditional statement:

If a quadrilateral is a square, then it is a rectangle.

Write this conditional statement in English using

the phrase “is sufficient for”

Construct a truth table for each of the following statements:

$P \wedge \neg Q$

Do any of these statements have the same truth table?

Construct a truth table for each of the following statements:

$\neg \left( {P \wedge Q} \right)$

Do any of these statements have the same truth table?