집합과 명제 전체 문제

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?

$P$

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?

$P \wedge Q$

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?

$P \vee Q$

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

$\neg P \to Q$

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

$Q \to P$

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

$P \vee Q$

Suppose that $P$ and $Q$ are statements for which $Q$ is false and $\neg P \to Q$ is true (and it is not known if $R$ is true or false). What conclusion (if any) can be made about the truth value of each of the following statements?

$\neg Q \to P$

Suppose that $P$ and $Q$ are statements for which $Q$ is false and $\neg P \to Q$ is true (and it is not known if $R$ is true or false). What conclusion (if any) can be made about the truth value of each of the following statements?

$P$

Suppose that $P$ and $Q$ are statements for which $Q$ is false and $\neg P \to Q$ is true (and it is not known if $R$ is true or false). What conclusion (if any) can be made about the truth value of each of the following statements?

$P \wedge R$

Suppose that $P$ and $Q$ are statements for which $Q$ is false and $\neg P \to Q$ is true (and it is not known if $R$ is true or false). What conclusion (if any) can be made about the truth value of each of the following statements?

$R \to \neg P$

Construct a truth table for each of the following statements:

$P \to Q$

Construct a truth table for each of the following statements:

$Q \to P$

Construct a truth table for each of the following statements:

$\neg P \to \neg Q$