명제와 조건

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”

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?

Construct a truth table for each of the following statements:

$\neg P \wedge \neg Q$

Do any of these statements have the same truth table?

Construct a truth table for each of the following statements:

$\neg P \vee \neg Q$

Do any of these statements have the same truth table?

Complete a truth table for $\left[ {\left( {Q \to P} \right) \wedge \left( {P \to Q} \right)} \right]$. Use the following columns: $P$, $Q$, $Q \to P$, $P \to Q$, and $\left[ {\left( {Q \to P} \right) \wedge \left( {P \to Q} \right)} \right]$. The last column of this table will be the truth table for $P \leftrightarrow Q$.

For statements $P$ and $Q$:

Use a truth table to show that $\left( {P \vee \neg P} \right)$ is a tautology.

For statements $P$ and $Q$:

Use a truth table to show that $\left( {P \wedge \neg P} \right)$ is a contradiction.

For statements $P$ and $Q$:

Use a truth table to determine if $P \to (P \vee Q)$ is a tautology, a contradiction, or neither.

Suppose that Daisy says, “If it does not rain, then I will play golf.” Later in the day you come to know that it did rain but Daisy still played golf. Was Daisy's statement true or false? Support your conclusion.

Suppose that $P$ and $Q$ are statements for which $P \to Q$ is true and for which $\neg Q$ is true. What conclusion (if any) can be made about the truth value of each of the following statements?

$P$